Gear dynamics analysis method considering uncertainty of modification parameters

Abstract

This invention relates to a gear dynamics analysis method considering the uncertainty of modification parameters. The method first constructs a deterministic dynamic model of the modified gear, then determines the center point, radius, and correlation coefficient of each uncertainty parameter. Based on the calculation of the standard first-order Chebyshev interpolation points and the first and second-order sampling points of the uncertainty vector, it constructs a global approximate function of the modified gear's dynamic response based on a low-order Chebyshev polynomial. Finally, it determines the upper and lower bounds of the modified gear's dynamic response and the correlation coefficient between the modified gear's dynamic response, achieving the goal of accurately and efficiently analyzing the dynamic response of the modified gear with relevant uncertainties.

Description

Technical Field

[0001] This invention relates to the field of gear dynamics analysis, and in particular to a gear dynamics analysis method that takes into account the uncertainty of modification parameters. Background Technology

[0002] Determining the dynamic response of modified gears is an important technical direction in the field of gear dynamics analysis. Many technicians have proposed methods for solving and determining the dynamic response of modified gears under deterministic conditions. However, there are generally uncertain parameters in gear transmission systems due to manufacturing errors, assembly errors, wear, and faults. These uncertain parameters lead to the fact that the actual dynamic response of modified gears is not a fixed value, but also has uncertainty.

[0003] To describe this uncertainty, three methods are commonly used to determine the uncertain dynamic response of modified gears: 1. Probabilistic method: This method requires determining the probability density function of the uncertainty parameter through a large number of repeated experiments, which is costly and difficult to accept; 2. Fuzzy method: This method requires describing the uncertainty parameter with a suitable membership function, which is too subjective and lacks universality in engineering practice; 3. Interval method: This method only requires the upper and lower bounds of the uncertainty parameter and does not require knowing the specific probability density of the uncertainty parameter to perform the calculation, and is gradually being accepted by engineering practice.

[0004] However, the interval method assumes that the uncertain parameters are independent and do not affect each other, which is inconsistent with the general law that uncertain parameters influence each other in reality. Therefore, it is necessary to introduce the concept of related uncertainty in the dynamic analysis of modified gears. Existing methods for determining the dynamic response of modified gears with related uncertainties require the construction of first-order Chebyshev polynomials or higher-order Chebyshev polynomials, each of which has its drawbacks: the former can only determine the upper and lower bounds of the dynamic response of modified gears and cannot describe the correlation between the dynamic responses of two modified gears; the latter, although it can describe the correlation between the dynamic responses of two modified gears, is prone to the "curse of dimensionality" and unacceptable computation time when dealing with a large number of uncertain parameters. Summary of the Invention

[0005] The purpose of this invention is to propose a gear dynamics analysis method and system that considers the uncertainty of the modification parameters. Under the premise of ensuring calculation accuracy, it realizes the function of calculating the upper and lower bounds of the dynamic response of the modified gear with relevant uncertainties and the correlation coefficient between the responses, and improves the calculation efficiency of the process of determining the dynamic response of the modified gear with relevant uncertainties as much as possible.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A gear dynamics analysis method considering the uncertainty of modification parameters includes the following steps:

[0008] S1: Obtain the deterministic design parameters of the modified gears and calculate the modification value E for each of the two pairs of meshing teeth p. p Where p = 1, 2; and the modification value E of the two pairs of meshing teeth. p The smaller value is taken as the static transmission error without load, and a set of dynamic equations for the modified gear is constructed to form a deterministic dynamic model of the modified gear;

[0009] S2: Determine the center point, radius, and correlation coefficient of each uncertainty parameter of the modified gear, and construct the uncertainty vector X;

[0010] S3: Substitute the center point of the uncertainty vector X into the aforementioned dynamic equations of the modified gear to solve for the dynamic transmission error of the modified gear, and further calculate the dynamic response of each modified gear. Where i = 1, 2, ..., m, m represents the number of dynamic responses of the modified gear;

[0011] S4: Calculate the standard first-order Chebyshev interpolation points Where i s =1,2,…,r, where r is the order of the Chebyshev polynomial;

[0012] S5: According to the obtained standard first-order Chebyshev interpolation points Construct the standard first-order Chebyshev vector and standard second-order Chebyshev vector And calculate the first-order sampling points of the uncertainty vector X. and second-order sampling points Where i s =1,2,…,r,i q =1,2,…,r;

[0013] S6: Sample the first-order points of the uncertainty vector described in step S5. and second-order sampling points Substituting the equations into the dynamic equations of the modified gear described above, we can obtain the corresponding dynamic responses of the modified gear. and where i=1,2…,m,i s =1,2,…,r,i q =1,2,…,r;

[0014] S7: Based on the solution value of the dynamic response of each modified gear in S3 And the corresponding solution value in S6 and Determine the dynamic response Y of each modified gear i The final global approximation function G i , i = 1, 2, ..., m;

[0015] S8: Based on the dynamic response Y of each modified gear obtained in step S7 i The final global approximation function G i Determine the dynamic response Y of each modified gear. i The upper and lower bounds of the dynamic response Y of any two modified gears. i and Y j The correlation coefficient θ between them ij , i = 1, 2, ..., m, j = 1, 2, ..., m, are used as the criteria for judging the best and worst performance of the vibration of the modified gear.

[0016] Preferably, the no-load static transfer error nlste in step S1 is calculated according to formula (1):

[0017] nlste=min{E1,E2} (1)

[0018] In the formula: E1 is the modification value of meshing tooth pair 1, and E2 is the modification value of meshing tooth pair 2.

[0019] Preferably, in step S1, the constructed set of dynamic equations for the modified gear is as follows:

[0020]

[0021]

[0022]

[0023]

[0024] Where t represents time, M1, k1, c1, I1, T1, m1, z1, α1, y1(t), and φ1(t) represent the mass, support stiffness, support damping, moment of inertia, input torque, module, number of teeth, pressure angle, vibration displacement along the meshing line at time t, and rotation angle at time t, respectively. M2, k2, c2, I2, T2, m2, z2, α2, y2(t), and φ2(t) represent the mass, support stiffness, support damping, moment of inertia, output torque, module, number of teeth, pressure angle, vibration displacement along the meshing line at time t, and rotation angle at time t, respectively. mp (t) represents the contact force between the meshing teeth and p, where p = 1, 2.

[0025] Preferably, in step S2, the uncertainty vector is represented as X = (x1, x2, ..., x...).n ) T , where n is the number of uncertainty parameters, and the relevant uncertainty of X is expressed as shown in equations (6) and (7):

[0026] x s =[x sL ,x sU ] = [x sC -x sR ,x sC +x sR (6)

[0027]

[0028] Where: s = 1, 2, ..., n, q = 1, 2, ..., n, x sL x sU x sC x sR x represents s The upper bound, lower bound, center point, and radius of ρ sq Let x be two uncertain parameters in X. s With x q The correlation coefficient between them.

[0029] Preferably, the standard first-order Chebyshev interpolation point in step S4 Calculate according to formula (8):

[0030]

[0031] Among them, i s =1,2,…,r, where r is the order of the Chebyshev polynomial.

[0032] Preferably, in step S5, the low-order sampling points of the uncertainty vector include first-order sampling points. and second-order sampling points Calculate according to equations (9) and (10) respectively:

[0033]

[0034]

[0035] Among them, i s =1,2,…,r,i q =1,2,…,r,X C For n center points x sC The vector X formed C =(x 1c x 2c , ..., x nc Standard first-order Chebyshev vector and standard second-order Chebyshev vector Calculate according to equations (11) and (12) respectively:

[0036]

[0037]

[0038] Where: standard first-order Chebyshev vector The s-th element is All other elements are 0, and the vector length is n; a standard second-order Chebyshev vector. The s-th and q-th elements are respectively and All other elements are 0, and the vector length is n.

[0039] R X For the radius x sR The resulting n×n dimensional matrix is ​​calculated according to equation (13):

[0040]

[0041] Φ is the result of the correlation coefficient ρ sq The resulting n×n dimensional matrix satisfies equation (14):

[0042]

[0043] Preferably, in step S7, Y i Global approximation function G i (ξ) is calculated according to formula (15):

[0044]

[0045] in, For the dynamic response Y of the modified gear i The univariate approximation function, the coefficient vector of which is derived from the dynamic response of the modified gear. Calibration yields that the independent variable in the univariate approximation function is ξ; For the dynamic response Y of the modified gear i The two-variable approximation function, where the coefficient vector is derived from the dynamic response of the modified gear. After calibration, the independent variable in the two-variable approximation function is ξ; i = 1, 2, ..., m, s = 1, 2, ..., n, q = 1, 2, ..., n.

[0046] Preferably, in step S8, the dynamic response Y of the modified gear... i The upper bound (Y) i ) U and lower bound (Y)i ) L The results are obtained by solving equations (16) and (17) using a genetic algorithm:

[0047]

[0048]

[0049] Where ξ=(ξ1,ξ2…,ξ) n ) T Ω ξ The standard uncertainty region is as shown in equation (18):

[0050] Ω ξ ={ξ|ξ T ξ≤1} (18)

[0051] The dynamic response Y of the modified gear is calculated according to equation (19). i and Y j The correlation coefficient θ between them ij :

[0052]

[0053] Where: i = 1, 2, ..., m, j = 1, 2, ..., m; Cov ξ =ΦΦ T Trace() represents the trace of a matrix. β i Calculate according to formula (20):

[0054]

[0055] α j and β j The calculation formulas are respectively with α i and β i same.

[0056] Preferably, the contact force F between the meshing teeth p is... mp (t) is calculated according to formula (21):

[0057]

[0058] Where, k mp (t) represents the meshing stiffness of the meshing tooth pair p at time t, and its calculation formula is shown in equation (22):

[0059]

[0060] Where, k cp (t), k ap (t), k bp (t), ksp (t) represents the contact stiffness, axial compressive stiffness, bending stiffness, and shear stiffness of the meshing tooth pair p at time t, respectively.

[0061] g p (t) represents the elastic deformation of the meshing tooth pair p at time t, and its calculation formula is shown in equation (23):

[0062] g p (t)=Δ(t)-(b1+b2)-max{0,E2-E1} (23)

[0063] Wherein, b1 and b2 are the tooth backlashes of the input gear and the output gear, respectively;

[0064] c mp (t) represents the meshing damping of the meshing tooth pair p at time t, and its calculation formula is shown in equation (24):

[0065]

[0066] Where ζ is the damping ratio;

[0067] Δ(t) is the relative displacement at time t, and its calculation formula is shown in equation (25):

[0068]

[0069] Preferably, the modified gear dynamic response Y i Univariate approximation function and two-variable approximation functions Calculate according to equations (26) and (27) respectively:

[0070]

[0071]

[0072] The independent variable ξ contains n independent variable parameters ξ1, ξ2, ..., ξ n ξ=(ξ1,ξ2…,ξ) n ) T coefficient vector according to The coefficient vector is obtained by fitting using the least squares method. according to The coefficient vector is obtained by fitting using the least squares method. and Calculate according to equations (28) and (29) respectively:

[0073]

[0074]

[0075] The beneficial effects of this invention are as follows: This invention provides a new approach to gear dynamics analysis considering the uncertainty of modification parameters. Compared to traditional probabilistic and fuzzy methods, it can estimate the upper and lower bounds of the gear dynamic response interval without prior knowledge of the probability density distribution of the uncertain parameters. Compared to existing gear dynamics analysis methods and systems considering the uncertainty of modification parameters, it can consider the correlation between uncertain parameters, thus broadening the applicability of the method. More importantly, the low-order Chebyshev polynomials used can significantly reduce the number of solutions to the deterministic modified gear dynamics model without reducing computational accuracy, thereby effectively shortening computation time, improving the design and analysis efficiency of modified gears, and making it possible to calculate the correlation coefficients between uncertain dynamic responses of modified gears. Attached Figure Description

[0076] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0077] Figure 1 This invention provides a method and system flowchart for gear dynamics analysis considering the uncertainty of profile modification parameters. Detailed Implementation

[0078] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0079] like Figure 1 As shown, in a preferred embodiment of the present invention, a gear dynamics analysis method considering the uncertainty of modification parameters is provided, the specific steps of which are as follows:

[0080] S1: Constructing a deterministic dynamic model of the modified gear, the specific steps of which include:

[0081] S1-1: Obtain the deterministic design parameters of the modified gear;

[0082] S1-2: Calculate the modification value E for each of the two pairs of meshing teeth p (where p = 1, 2). p ;

[0083] S1-3: Calculate the no-load static transfer error nlste;

[0084] In an embodiment of the present invention, the above-mentioned no-load static transfer error nlste is calculated by the following formula:

[0085] nlste=min{E1,E2}

[0086] In the formula: E1 is the modification value of meshing tooth pair 1, and E2 is the modification value of meshing tooth pair 2.

[0087] S1-4: Construct the dynamic equations for the modified gear.

[0088] In an embodiment of the present invention, the modified gear dynamics equations constructed above are as follows:

[0089]

[0090]

[0091]

[0092]

[0093] Where t represents time, M1, k1, c1, I1, T1, m1, z1, α1, y1(t), and φ1(t) represent the mass, support stiffness, support damping, moment of inertia, input torque, module, number of teeth, pressure angle, vibration displacement along the meshing line at time t, and rotation angle at time t, respectively. M2, k2, c2, I2, T2, m2, z2, α2, y2(t), and φ2(t) represent the mass, support stiffness, support damping, moment of inertia, output torque, module, number of teeth, pressure angle, vibration displacement along the meshing line at time t, and rotation angle at time t, respectively. mp (t) represents the contact force between the meshing teeth and p, where p = 1, 2.

[0094] In an embodiment of the present invention, the contact force F between the aforementioned meshing teeth pair p is... mp (t) is calculated using the following formula:

[0095]

[0096] Where, k mp (t) represents the meshing stiffness of the meshing tooth pair p at time t, and its calculation formula is shown in the following equation:

[0097]

[0098] Where, k cp (t), k ap (t), k bp (t), k sp (t) represents the contact stiffness, axial compressive stiffness, bending stiffness, and shear stiffness of the meshing tooth pair p at time t, respectively.

[0099] g p(t) represents the elastic deformation of the meshing tooth pair p at time t, and its calculation formula is shown in the following equation:

[0100] g p (t)=Δ(t)-(b1+b2)-max{0,E2-E1}

[0101] Wherein, b1 and b2 are the tooth backlashes of the input gear and the output gear, respectively;

[0102] c mp (t) represents the meshing damping of the meshing tooth pair p at time t, and its calculation formula is shown in the following equation:

[0103]

[0104] Where ζ is the damping ratio;

[0105] Δ(t) is the relative displacement at time t, and its calculation formula is shown below:

[0106]

[0107] S2: Determine the center point, radius, and correlation coefficient of each uncertainty parameter of the modified gear, and construct the uncertainty vector X.

[0108] S3: Substitute the center point of the uncertainty vector X into the aforementioned dynamic equations of the modified gear to solve for the dynamic transmission error of the modified gear, and further calculate the dynamic response of each modified gear. Where i = 1, 2, ..., m, m represents the number of dynamic responses of the modified gear.

[0109] In an embodiment of the present invention, in step S2 above, the uncertainty vector is represented as X = (x1, x2, ..., x...). n ) T Let n be the number of uncertainty parameters. The relevant uncertainties of X are expressed as follows:

[0110] x s =[x sL ,x sU ] = [x sC -x sR ,x sC +x sR ]

[0111]

[0112] Where: s = 1, 2, ..., n, q = 1, 2, ..., n, x sL x sU x sC x sR x represents sThe upper bound, lower bound, center point, and radius of ρ sq Let x be two uncertain parameters in X. s With x q The correlation coefficient between them.

[0113] S4: Calculate the standard first-order Chebyshev interpolation points Where i s =1,2,…,r, where r is the order of the Chebyshev polynomial.

[0114] In an embodiment of the present invention, the standard first-order Chebyshev interpolation point in step S4 above... Calculate using the following formula:

[0115]

[0116] Among them, i s =1,2,…,r, where r is the order of the Chebyshev polynomial.

[0117] S5: According to the obtained standard first-order Chebyshev interpolation points Construct the standard first-order Chebyshev vector and standard second-order Chebyshev vector And calculate the first-order sampling points of the uncertainty vector X. and second-order sampling points Where i s =1,2,…,r,i q =1,2,…,r.

[0118] In an embodiment of the present invention, in step S5 above, the low-order sampling points of the uncertainty vector include first-order sampling points. and second-order sampling points Calculate according to the following formulas:

[0119]

[0120]

[0121] Among them, i s =1,2,…,r,i q =1,2,…,r,X C For n center points x sC The vector X formed C =(x 1c x 2c , ..., x nc Standard first-order Chebyshev vector and standard second-order Chebyshev vector Calculate according to the following formulas:

[0122]

[0123]

[0124] Where: standard first-order Chebyshev vector The s-th element is All other elements are 0, and the vector length is n; a standard second-order Chebyshev vector. The s-th and q-th elements are respectively and All other elements are 0, and the vector length is n.

[0125] R X For the radius x sR The resulting n×n dimensional matrix is ​​calculated using the following formula:

[0126]

[0127] Φ is the result of the correlation coefficient ρ sq The resulting n×n dimensional matrix satisfies the following equation:

[0128]

[0129] S6: Sample the first-order points of the uncertainty vector described in step S5. and second-order sampling points Substituting the equations into the dynamic equations of the modified gear described above, we can obtain the corresponding dynamic responses of the modified gear. and where i=1,2…,m,i s =1,2,…,r,i q =1,2,…,r.

[0130] S7: Based on the solution value of the dynamic response of each modified gear in S3 And the corresponding solution value in S6 and Determine the dynamic response Y of each modified gear i The final global approximation function G i , i = 1, 2, ..., m.

[0131] In an embodiment of the present invention, in step S7 above, Y i Global approximation function G i (ξ) is calculated using the following formula:

[0132]

[0133] in, For the dynamic response Y of the modified gear iThe univariate approximation function, the coefficient vector of which is derived from the dynamic response of the modified gear. Calibration yields that the independent variable in the univariate approximation function is ξ; For the dynamic response Y of the modified gear i The two-variable approximation function, where the coefficient vector is derived from the dynamic response of the modified gear. After calibration, the independent variable in the two-variable approximation function is ξ; i = 1, 2, ..., m, s = 1, 2, ..., n, q = 1, 2, ..., n.

[0134] It should be noted that, in this invention, for each modified gear dynamic response Y i Steps S3 and S6 both obtain the dynamic response of the modified gear by solving the dynamic equations of the modified gear described in steps S1-4. (i = 1, 2, ..., m) and dynamic response of modified gears and (i = 1, 2, ..., m, i s i p =1,2…,r), but these three are all the final modified gear dynamic response Y. i The solution value needs to be obtained through the aforementioned global approximation function G. i (ξ) is used to obtain the final dynamic response Y of the modified gear. i (i = 1, 2, ..., m).

[0135] In an embodiment of the present invention, the above-mentioned modified gear dynamic response Y i Univariate approximation function and two-variable approximation functions Calculate according to the following formulas:

[0136]

[0137]

[0138] The vector-form independent variable ξ contains n independent variable parameters ξ1, ξ2, ..., ξ n ξ=(ξ1,ξ2…,ξ) n ) T coefficient vector according to The coefficient vector is obtained by fitting using the least squares method. according to The coefficient vector is obtained by fitting using the least squares method. and Calculate according to the following formulas:

[0139]

[0140]

[0141] S8: Based on the dynamic response Y of each modified gear obtained in step S7 i The final global approximation function G i Determine the dynamic response Y of each modified gear. i The upper and lower bounds of the dynamic response Y of any two modified gears. i and Y j The correlation coefficient θ between them ij , i = 1, 2, ..., m, j = 1, 2, ..., m, are used as the criteria for judging the best and worst performance of the vibration of the modified gear.

[0142] In an embodiment of the present invention, in step S8 above, the dynamic response Y of the modified gear... i The upper bound (Y) i ) U and lower bound (Y) i ) L The following two objective functions are obtained by solving them using a genetic algorithm:

[0143] (Y i ) U =maxG i (ξ)

[0144] stξ∈Ω ξ

[0145] (Y i ) L =minG i (ξ)

[0146] stξ∈Ω ξ

[0147] Where ξ=(ξ1,ξ2…,ξ) n ) T Ω ξ The standard uncertainty region is as shown in the following equation:

[0148] Ω ξ ={ξ|ξ T ξ≤1}

[0149] Calculate the dynamic response Y of the modified gear using the following formula i and Y j The correlation coefficient θ between them ij :

[0150]

[0151] Where: i = 1, 2, ..., m, j = 1, 2, ..., m; Cov ξ=ΦΦ T Trace() represents the trace of a matrix. β i Calculate using the following formula:

[0152]

[0153] α j and β j The calculation formulas are respectively with α i and β i same.

[0154] In this invention, the final result is the dynamic response Y of each modified gear in step S8. i The upper and lower bounds of the dynamic response Y of any two modified gears. i and Y j The correlation coefficient θ between them ij Based on the upper and lower bounds and correlation coefficients of these responses, the specific range of the dynamic response of the modified gear can be determined, thus providing a basis for evaluating the best and worst performance of the modified gear vibration.

[0155] It should be noted that the specific modified gear dynamic response Y that can be selected in this invention is... i The appropriate value can be selected based on the actual purpose of vibration assessment for modified gears. Below, this invention uses the peak-to-peak value of the dynamic transmission error of the modified gear and the root mean square value of the output gear support force as two exemplary examples of the dynamic response Y of the modified gear. i This is to demonstrate the specific implementation and technical effects of the present invention.

[0156] Example

[0157] In this embodiment, a gear dynamics analysis method considering the uncertainty of the modification parameters is provided, which uses the peak-to-peak dynamic transmission error PPDTE of the modified gear and the root mean square value of the output gear support force (F) as the basis for the analysis. y2 ) RMS As two exemplary examples, the dynamic response Y of the modified gear i This method is used to evaluate the best and worst performance of modified gear vibration. The specific steps of this method are as follows:

[0158] S1: Constructing a deterministic dynamic model of the modified gear, the specific steps of which include:

[0159] S1-1: Obtain the deterministic design parameters for the modified gears as shown in Table 1, including: the contact ratio ε and damping ratio ζ of each gear pair, and the number of teeth z of the two gears in this gear pair. i Modulus m i Tooth width B i Pressure angle α i displacement coefficient x iTooth flank clearance b i Quality M i Moment of inertia I i Support stiffness k i Support damping c i Rotational speed n i Torque T i , elastic modulus and Poisson's ratio. The subscripts "1" and "2" represent quantities related to the input and output gears, respectively.

[0160] Table 1

[0161]

[0162] S1-2: Let the tooth pairs meshing below the highest point of single tooth meshing be called meshing tooth pair 1, and the tooth pairs meshing above the highest point of single tooth meshing be called meshing tooth pair 2. The modification values ​​E1 and E2 of meshing tooth pair 1 and meshing tooth pair 2 are calculated according to Equation (1) and Equation (2) respectively:

[0163]

[0164]

[0165] Where a1, L1, and s1 are the modification amount, dimensionless modification length, and modification index of the input gear, respectively; a2, L2, and s2 are the modification amount, dimensionless modification length, and modification index of the output gear, respectively; and θ is the rolling angle corresponding to the meshing point of the input gear in meshing gear pair 1. A θ B θ LPSTC θ HPSTC Let θ represent the input gear rolling angles corresponding to the meshing start point, meshing end point, lowest point of single-tooth meshing, and highest point of single-tooth meshing, respectively, and calculate them according to formula (3):

[0166]

[0167] Here, mod() is the remainder operator.

[0168] S1-3: Using the modification value E of two pairs of meshing teeth p The smaller value is used to calculate the no-load static transfer error nlste, as shown in equation (4):

[0169] nlste=min{E1(θ),E2(θ)} (4)

[0170] S1-4: Write down the dynamic equations of the modified gear to form a deterministic dynamic model of the modified gear, as shown in equations (5) to (8):

[0171]

[0172]

[0173]

[0174]

[0175] Where y1(t), y2(t), φ1(t), and φ2(t) represent the vibration displacement of the input gear along the meshing line at time t, the vibration displacement of the output gear along the meshing line at time t, the rotation angle of the input gear at time t, and the rotation angle of the output gear at time t, respectively; M1, k1, c1, I1, T1, m1, z1, and α1 represent the mass, support stiffness, support damping, moment of inertia, input torque, module, number of teeth, and pressure angle of the input gear, respectively; and M2, k2, c2, I2, T2, m2, z2, and α2 represent the mass, support stiffness, support damping, moment of inertia, output torque, module, number of teeth, and pressure angle of the output gear, respectively. mp (t) represents the contact force between the meshing teeth and p (p = 1, 2), calculated according to equation (9):

[0176]

[0177] Where, k mp (t) is the meshing stiffness of the meshing tooth pair p at time t (p=1,2), calculated according to equation (10):

[0178]

[0179] Where, k cp (t), k ap (t), k bp (t), k sp (t) represents the contact stiffness, axial compressive stiffness, bending stiffness, and shear stiffness of the meshing tooth pair p at time t (p = 1, 2);

[0180] g p (t) represents the elastic deformation of the meshing tooth pair p at time t (p = 1, 2), calculated according to equation (11):

[0181] g p (t)=Δ(t)-(b1+b2)-max{0,E2-E1} (11)

[0182] c mp (t) represents the meshing damping of the meshing tooth pair p at time t (p = 1, 2), calculated according to equation (12):

[0183]

[0184] Δ(t) is the relative displacement at time t, calculated according to equation (13):

[0185]

[0186] S2: The modification amount of the input gear, the modification amount of the output gear, the dimensionless modification length of the input gear, the dimensionless modification length of the output gear, the modification index of the input gear, and the modification index of the output gear are taken as uncertainty parameters and denoted as x1, x2, x3, x4, x5, and x6 respectively. They are then uniformly written into the uncertainty vector X = (x1, x2, ..., x6). T The uncertainty of X is expressed as shown in Equation (14) and Table 2:

[0187] x s ∈x sI =[x sL ,x sU ] = [x sC -x sR ,x sC +x sR ],s=1,2…,6 (14)

[0188] Table 2

[0189]

[0190]

[0191] Where; x sI x represents s The upper and lower bounds of x sL x sU x sC x sR They represent x respectively s The upper bound, lower bound, center point, and radius of X are shown in Table 3 in this embodiment. Let x be the two uncertainty parameters in X. s With x q The correlation coefficient between them is ρ sq (s,q=1,2,…,6).

[0192] Table 3

[0193] <![CDATA[x 1U ]]> <![CDATA[x 2U ]]> <![CDATA[x 3U ]]> <![CDATA[x 4U ]]> <![CDATA[x 5U ]]> <![CDATA[x 6U ]]> 11μm 11μm 1.1 1.1 2.2 2.2 <![CDATA[x 1L ]]> <![CDATA[x 2L ]]> <![CDATA[x 3L ]]> <![CDATA[x 4L ]]> <![CDATA[x 5L ]]> <![CDATA[x 6L <!-- 13 -->]]> 9μm 9μm 0.9 0.9 1.8 1.8 <![CDATA[x 1C ]]> <![CDATA[x 2C ]]> <![CDATA[x 3C ]]> <![CDATA[x 4C ]]> <![CDATA[x 5C ]]> <![CDATA[x 6C ]]> 10μm 10μm 1 1 2 2 <![CDATA[x 1R ]]> <![CDATA[x 2R ]]> <![CDATA[x 3R ]]> <![CDATA[x 4R ]]> <![CDATA[x 5R ]]> <![CDATA[x 6R ]]> 1μm 1μm 0.1 0.1 0.2 0.2

[0194] S3: Substitute the center point of the uncertainty vector X into the aforementioned dynamic equations of the modified gear to solve for the dynamic transmission error of the modified gear, and further calculate the dynamic response of each modified gear. In this embodiment, let X = X C =(x 1C ,x 2C …,x6C ) T Substituting these equations (1) to (13) into the deterministic dynamic model of the modified gear, and solving it using the fourth-order Runge-Kutta method, the dynamic transmission error of the modified gear is obtained as shown in equation (15):

[0195]

[0196] Based on the dynamic transmission error DTE(t) of the modified gears mentioned above, the dynamic response of m=2 modified gears in this embodiment can be calculated. The specific calculations are as follows:

[0197] Calculate the dynamic response of the first modified gear The peak-to-peak value of the dynamic transmission error of the modified gear is shown in equation (16):

[0198] PPDTE C =max{DTE(t)}-min{DTE(t)} (16)

[0199] The output gear support force is calculated as shown in equation (17):

[0200]

[0201] Calculate the dynamic response of the second modified gear The root mean square value of the output gear support force is shown in equation (18):

[0202]

[0203] Where T = 3s.

[0204] S4: Calculate the standard first-order Chebyshev interpolation points according to equation (19):

[0205]

[0206] Among them, i s =1,2,3,4,j =1,2…,6.

[0207] S5: According to the obtained standard first-order Chebyshev interpolation points Construct the standard first-order Chebyshev vectors according to equations (20) and (21) respectively. and standard second-order Chebyshev vector

[0208]

[0209]

[0210] Among them, i s i q=1,2,3,4; In equation (20), the standard first-order Chebyshev vector The s-th element is The remaining elements are all 0, and the vector length is n = 6; in equation (21), the standard second-order Chebyshev vector The s-th and q-th elements are respectively and All other elements are 0, and the vector length is n = 6.

[0211] Based on the standard first-order Chebyshev vector and standard second-order Chebyshev vector The first-order sampling points of the uncertainty vector X are reconstructed according to equations (22) and (23) respectively. and second-order sampling points Where i s =1,2…,4,i q =1,2…,4:

[0212]

[0213]

[0214] Among them, X C For n = 6 center points x sC The vector X formed C =(x 1c x 2c , ..., x nc ); R X For the radius x sR The resulting 6×6 dimensional matrix is ​​calculated according to equation (24):

[0215]

[0216] Φ is the result of the correlation coefficient ρ sq The resulting 6×6 dimensional matrix can be determined through correlation analysis of the measured data and satisfies equation (25):

[0217]

[0218] S6: Order Substituting these equations into the deterministic dynamic model of the modified gear constructed from equations (1) to (13), and solving it using the fourth-order Runge-Kutta method, the peak-to-peak value of the dynamic transmission error of the modified gear corresponding to the first-order sampling point of the uncertainty vector is calculated according to equations (15) to (18). and the root mean square value of the output gear support force make Substituting these equations into the deterministic dynamic model of the modified gear constructed from equations (1) to (13), and solving it using the fourth-order Runge-Kutta method, the peak-to-peak value of the dynamic transmission error of the modified gear corresponding to the second-order sampling points of the uncertainty vector is calculated according to equations (15) to (18). and

[0219] S7: Based on the solution values ​​of the dynamic response of each modified gear in S3 and the solution values ​​obtained from the first-order sampling points and second-order sampling points in S6, determine the final overall approximate function of the dynamic response of each modified gear.

[0220] In this embodiment, a single-variable approximation function for the peak-to-peak value of the dynamic transmission error of the modified gear and the root mean square value of the output gear support force is constructed, as shown in equations (26) and (27):

[0221]

[0222]

[0223] Among them, ξ=(ξ1,ξ2…,ξ6) T coefficient vector and According to respectively and The coefficient vector (1, 2, 3, 4) was obtained by fitting using the least squares method. Calculate according to formula (28):

[0224]

[0225] Simultaneously, a two-variable approximation function of the peak-to-peak value of the transmission error of the modified gear and the root mean square value of the support force of the output gear, in the form of equations (29) and (30), is constructed:

[0226]

[0227]

[0228] The independent variable ξ contains six independent variable parameters ξ1, ξ2, ..., ξ6: ξ = (ξ1, ξ2, ..., ξ6) T coefficient vector and According to respectively and (i s i q The coefficient vector (1, 2, 3, 4) was obtained by fitting using the least squares method. Calculate according to formula (31):

[0229]

[0230] The overall approximate functions G for the peak-to-peak value of the transmission error of the modified gear and the root mean square value of the support force of the output gear are calculated according to equations (32) and (33), respectively. 1 (ξ) and G 2 (ξ):

[0231]

[0232]

[0233] S8: Using a genetic algorithm to solve equations (34) and (35), the upper bound PPDTE of the peak-to-peak value of the dynamic transmission error of the modified gear is obtained. U and the lower PPDTE L And the upper bound of the root mean square value of the output gear support force. and the lower realm

[0234]

[0235]

[0236] Among them, Ω ξ The standard uncertainty region is of the form (36):

[0237]

[0238] The correlation coefficient between the peak-to-peak value of the dynamic transmission error of the modified gear and the root mean square value of the output gear support force is calculated according to formula (37):

[0239]

[0240] Where Trace() represents the trace of the matrix. β i (i=1,2) Calculate according to formula (38):

[0241]

[0242] α j and β j The calculation formulas are respectively with α i and β i The same applies; simply change the superscript i to j for each parameter in equations (37) and (38).

[0243] Based on the upper and lower bounds and correlation coefficients of the peak-to-peak value of the dynamic transmission error of the modified gear and the root mean square value of the output gear support force, the specific range of the dynamic response of the modified gear can be clearly defined, thus providing a basis for evaluating the best and worst performance of the modified gear vibration. For the root mean square value of the output gear support force and the peak-to-peak value of the dynamic transmission error of the modified gear, their respective upper bounds represent the maximum possible amplitude of gear vibration, i.e., the worst performance of the gear during operation; conversely, their respective lower bounds represent the minimum possible amplitude of gear vibration, i.e., the best performance of the gear during operation. As for the correlation coefficient, it affects the uncertainty domain formed by the two responses. The closer the correlation coefficient is to 0, the larger the uncertainty domain of the response, indicating higher vibration uncertainty, which is detrimental to the stable operation of the gear transmission system; conversely, a higher correlation coefficient indicates lower uncertainty in the gear transmission system, which is beneficial to the stable operation of the gear transmission system.

[0244] Table 4 compares the upper and lower bounds of the peak-to-peak value of the dynamic transmission error of the modified gear and the root mean square value of the supporting force, as well as the correlation coefficients between them, calculated by Monte Carlo simulation, the higher-order Chebyshev polynomial method, and the gear dynamics analysis method considering the uncertainty of the modification parameters described in this invention. Among these, Monte Carlo simulation, by determining the upper and lower bounds of the uncertain dynamic response of the modified gear and the correlation coefficients between them through extensive sampling, yields the most accurate results. The results calculated by the gear dynamics analysis method considering the uncertainty of the modification parameters described in this invention and the higher-order Chebyshev polynomial method are similar, with relative errors between them and the results obtained by Monte Carlo simulation all within 5%, verifying the calculation accuracy of this invention. Furthermore, the gear dynamics analysis method considering the uncertainty of the modification parameters described in this invention requires far fewer iterations to solve for the deterministic dynamic response of the modified gear than Monte Carlo simulation and the higher-order Chebyshev polynomial method, verifying the computational efficiency of the gear dynamics analysis method considering the uncertainty of the modification parameters described in this invention. In summary, the gear dynamics analysis method considering the uncertainty of the modification parameters described in this invention can calculate the upper and lower bounds of the dynamic response of the modified gear with relevant uncertainties and the correlation coefficient between the responses, while ensuring the accuracy of the calculation. It also effectively improves the calculation efficiency of the process of determining the dynamic response of the modified gear with relevant uncertainties.

[0245] Table 4

[0246]

[0247] It should be noted that, based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without inventive effort are within the scope of protection of this invention. Furthermore, unless otherwise stated, the technical or scientific terms used in this application should have the ordinary meaning understood by those skilled in the art.

Claims

1. A gear dynamics analysis method considering the uncertainty of modification parameters, characterized in that, Includes the following steps: S1: Obtain the deterministic design parameters of the modified gears and calculate the modification value E for each of the two pairs of meshing teeth p. p Where p = 1, 2; and the modification value E of the two pairs of meshing teeth. p The smaller value is taken as the static transmission error without load, and a set of dynamic equations for the modified gear is constructed to form a deterministic dynamic model of the modified gear; S2: Determine the center point, radius, and correlation coefficient of each uncertainty parameter of the modified gear, and construct the uncertainty vector X; S3: Substitute the center point of the uncertainty vector X into the aforementioned dynamic equations of the modified gear to solve for the dynamic transmission error of the modified gear, and further calculate the dynamic response of each modified gear. Where i = 1, 2, ..., m, m represents the number of dynamic responses of the modified gear; S4: Calculate the standard first-order Chebyshev interpolation points Where i s =1,2,…,r, where r is the order of the Chebyshev polynomial; S5: According to the obtained standard first-order Chebyshev interpolation points Construct the standard first-order Chebyshev vector and standard second-order Chebyshev vector And calculate the first-order sampling point X of the uncertainty vector X. is and second-order sampling points Where i s =1,2,…,r,i q =1,2,…,r; S6: Sample the first-order points of the uncertainty vector described in step S5. and second-order sampling points Substituting the equations into the dynamic equations of the modified gear described above, we can obtain the corresponding dynamic responses of the modified gear. and where i=1,2…,m,i s =1,2,…,r,i q =1,2,…,r; S7: Based on the solution value of the dynamic response of each modified gear in S3 And the corresponding solution value in S6 and Determine the dynamic response Y of each modified gear i The final global approximation function G i , i = 1, 2, ..., m; S8: Based on the dynamic response Y of each modified gear obtained in step S7 i The final global approximation function G i Determine the dynamic response Y of each modified gear. i The upper and lower bounds of the dynamic response Y of any two modified gears. i and Y j The correlation coefficient θ between them ij , i = 1, 2, ..., m, j = 1, 2, ..., m, are used as the criteria for judging the best and worst performance of the vibration of the modified gear.

2. The gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 1, characterized in that, The no-load static transfer error nlste in step S1 is calculated using the following formula: nlste=min{E1,E2} In the formula: E1 is the modification value of meshing tooth pair 1, and E2 is the modification value of meshing tooth pair 2.

3. The gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 1, characterized in that, In step S1, the constructed set of dynamic equations for the modified gear is as follows: Where t represents time, M1, k1, c1, I1, T1, m1, z1, α1, y1(t), and φ1(t) represent the mass, support stiffness, support damping, moment of inertia, input torque, module, number of teeth, pressure angle, vibration displacement along the meshing line at time t, and rotation angle at time t, respectively. M2, k2, c2, I2, T2, m2, z2, α2, y2(t), and φ2(t) represent the mass, support stiffness, support damping, moment of inertia, output torque, module, number of teeth, pressure angle, vibration displacement along the meshing line at time t, and rotation angle at time t, respectively. mp (t) represents the contact force between the meshing teeth and p, where p = 1, 2.

4. The gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 1, characterized in that, In step S2, the uncertainty vector is represented as X = (x1, x2, ..., x...). n ) T Let n be the number of uncertainty parameters. The relevant uncertainties of X are expressed as follows: x s =[x sL ,x sU ]=[x sC -x sR ,x sC +x sR ] Where: s = 1, 2, ..., n, q = 1, 2, ..., n, x sL x sU x sC x sR x represents s The upper bound, lower bound, center point, and radius of ρ sq Let x be two uncertain parameters in X. s With x q The correlation coefficient between them.

5. A gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 1, characterized in that, The standard first-order Chebyshev interpolation point in step S4 Calculate using the following formula: Among them, i s =1,2,…,r, where r is the order of the Chebyshev polynomial.

6. The gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 1, characterized in that, In step S5, the low-order sampling points of the uncertainty vector include first-order sampling points. and second-order sampling points Calculate according to the following formulas: Among them, i s =1,2,…,r,i q =1,2,…,r,X C For n center points x sC The vector X formed C =(x 1c x 2c , ..., x nc Standard first-order Chebyshev vector and standard second-order Chebyshev vector Calculate according to the following formulas: Where: standard first-order Chebyshev vector The s-th element is All other elements are 0, and the vector length is n; a standard second-order Chebyshev vector. The s-th and q-th elements are respectively and All other elements are 0, and the vector length is n. R X For the radius x sR The resulting n×n dimensional matrix is ​​calculated using the following formula: Φ is the result of the correlation coefficient ρ sq The resulting n×n dimensional matrix satisfies the following equation:

7. A gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 1, characterized in that, In step S7, Y i Global approximation function G i (ξ) is calculated using the following formula: in, For the dynamic response Y of the modified gear i The univariate approximation function, the coefficient vector of which is derived from the dynamic response of the modified gear. Calibration yields that the independent variable in the univariate approximation function is ξ; For the dynamic response Y of the modified gear i The two-variable approximation function, where the coefficient vector is derived from the dynamic response of the modified gear. After calibration, the independent variable in the two-variable approximation function is ξ; i = 1, 2, ..., m, s = 1, 2, ..., n, q = 1, 2, ..., n.

8. The gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 1, characterized in that, In step S8, the dynamic response Y of the modified gear i The upper bound (Y) i ) U and lower bound (Y) i ) L The following two objective functions are obtained by solving them using a genetic algorithm: (Y i ) U <maxG i (ξ) s.t.ξ∈Ω ξ (Y i ) L =minG i (ξ) s.t.ξ∈Ω ξ Where ξ=(ξ1,ξ2…,ξ n ) T Ω ξ The standard uncertainty region is as shown in the following equation: Oh ξ ={ξ|ξ T ξ≤1} Calculate the dynamic response Y of the modified gear using the following formula i and Y j The correlation coefficient θ between them ij : Where: i = 1, 2, ..., m, j = 1, 2, ..., m; Cov ξ =ΦΦ T Trace() represents the trace of a matrix. β i Calculate using the following formula: α j and β j The calculation formulas are respectively with α i and β i same.

9. A gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 3, characterized in that, The contact force F between the meshing teeth p mp (t) is calculated using the following formula: Where, k mp (t) represents the meshing stiffness of the meshing tooth pair p at time t, and its calculation formula is shown in the following equation: Where, k cp (t), k ap (t), k bp (t), k sp (t) represents the contact stiffness, axial compressive stiffness, bending stiffness, and shear stiffness of the meshing tooth pair p at time t, respectively. g p (t) represents the elastic deformation of the meshing tooth pair p at time t, and its calculation formula is shown in the following equation: g p (t)=Δ(t)-(b1+b2)-max{0,E2-E1} Wherein, b1 and b2 are the tooth backlashes of the input gear and the output gear, respectively; c mp (t) represents the meshing damping of the meshing tooth pair p at time t, and its calculation formula is shown in the following equation: Where ζ is the damping ratio; Δ(t) is the relative displacement at time t, and its calculation formula is shown below:

10. A gear dynamics analysis method considering the uncertainty of modification parameters as described in claim 7, characterized in that, The dynamic response Y of the modified gear i Univariate approximation function and two-variable approximation functions Calculate according to the following formulas: The independent variable ξ contains n independent variable parameters ξ1, ξ2, ..., ξ n ξ=(ξ1,ξ2…,ξ) n ) T coefficient vector according to The coefficient vector is obtained by fitting using the least squares method. according to The coefficient vector is obtained by fitting using the least squares method. and Calculate according to the following formulas:

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