**Introduction**

Spiral bevel gears are widely used in various fields such as automobiles, aviation, and mining machinery due to their high coincidence degree, strong load-carrying capacity, high transmission ratio, and high transmission efficiency. The manufacturing accuracy of spiral bevel gears is directly related to the errors of CNC machine tools. Machine tool errors include geometric errors, thermal errors, and servo control errors, among which machine tool geometric errors are repeatable and stable and can be accurately compensated through the CNC system. Therefore, studying the characteristics of machine tool geometric errors and determining the key geometric error items is of great importance for tooth surface deviation compensation and the allocation and maintenance of machine tool motion accuracy.

**Sensitivity Analysis Methods**

There are mainly two types of sensitivity analysis methods: local sensitivity analysis methods represented by differential method, difference method, and perturbation method, and global sensitivity analysis methods represented by regression analysis method, variance method, and screening method. Through sensitivity analysis, the key input factors and non-key input factors can be determined. In the actual analysis and calculation, the non-key input factors can be ignored first, which can significantly reduce the complexity of the system and the difficulty of analysis, and greatly reduce the calculation amount and processing difficulty of relevant data.

**Local Sensitivity Analysis Method**

Local sensitivity analysis is a kind of single-factor analysis method. By making a small change to a single input factor each time while keeping all other factors unchanged, the sensitivity coefficient is determined by the differential of the output result to the input parameter or the change of a single input factor to the output result. The local sensitivity concept is clear and the calculation is simple, which is mainly suitable for linear models and models with weak nonlinearity.

Step | Description |
---|---|

Start | Begin the analysis process. |

Input the change amount of the first parameter | Specify the variation in the first input parameter. |

Calculate the theoretical output result | Determine the expected output based on the model and the changed parameter. |

Calculate the actual output result | Obtain the actual output considering all factors. |

Calculate the deviation value | Find the difference between the theoretical and actual output results. |

i = i + 1 | Move to the next parameter. |

Output all deviation values | Present the deviations for all parameters. |

End | Finish the analysis. |

**Sobol Global Sensitivity Analysis**

The global sensitivity analysis method not only considers the range and distribution law of each parameter but also takes all parameters into account during the analysis and calculation, considering the influence of the mutual coupling of each input parameter on the result. However, the global sensitivity analysis method generally requires a certain scale of sampling to obtain the input parameters of the model, and then performs sensitivity analysis. Especially for the case with more parameters, the calculation amount will be relatively large.

Taking the Sobol method as an example, it is a global sensitivity method based on variance. This method mainly calculates the sensitivity coefficient of the input parameter by decomposing the variance of the model into sub-items.

Suppose there are n input parameters, denoted by x_i (i = 1, 2,…, n). The model is represented by the multivariable function K = f(X), where X = (x_1, x_2,…, x_n). The function f(X) can be decomposed as follows:

f(X) = f_0 + ∑_(i=1)^n f_i(x_i) + ∑_(i<j)^n f_i,j(x_i, x_j) +… + f_1,…,n(x_1, x_2,…, x_n)

Here, f_0 is the expected value of the output result f(X) calculated by the input parameters, which is a constant; f_i(x_i) represents the output result under the sole action of the input parameter x_i; f_i,j(x_i, x_j) represents the output result under the joint action of the input parameters x_i and x_j; f_1,…,n(x_1, x_2,…, x_n) represents the output result under the joint action of all input parameters; and the same applies to other orders.

The total variance D can be expressed as:

D = ∫_(0)^1… ∫_(0)^1 f^2(X) dX – f_0^2

The variance components can be obtained as follows:

D_(i_1,…,i_s) = ∫_(0)^1… ∫_(0)^1 f_(i_1,…,i_s)^2(x_(i_1),…, x_(i_s)) dx_(i_1)… dx_(i_s) (where 1 ≤ s ≤ n)

By squaring and integrating both sides of equation (1) according to the Sobol method, we get:

D = ∑_(i=1)^n D_i + ∑_(i<j)^n D_i,j +… + D_1,2,…,n

The sensitivity coefficient can be obtained through the following formula:

S_(i_1,…,i_k) = D_(i_1,…,i_s) / D

Since the variance is non-negative, S_(i_2,…,i_s) is also non-negative. According to equations (4) and (5), we have:

∑_(i=1)^n S_i + ∑_(i<j)^n S_i,j +… + S_1,2,…,n = 1

Here, S_i is the first-order sensitivity coefficient corresponding to the variable x_i; S_i,j (i < j) is the second-order sensitivity coefficient corresponding to the mutual coupling effect of the variables x_i and x_j. Similarly, the sensitivity coefficient of each order can be obtained. The global sensitivity coefficient S_(i)^tot corresponding to the variable x_i can be obtained by summing up the sensitivity coefficients related to the variable x_i. The calculation method is as follows:

S_(i)^tot = S_i + ∑_(i=k)^s S_i,k +… + S_1,2,…,n (where the positive integer k takes values in the range 1 ≤ k ≤ n and k ≠ i)

**Monte Carlo Estimation**

In the process of solving the sensitivity coefficient using the Sobol method, it involves the solution of multiple integrals, which is usually very difficult for complex models. Therefore, the Monte Carlo method is often used to approximately simulate the solution of multiple integrals. The general calculation method is as follows: perform two independent samplings on the input parameters to obtain two independent sampling matrices E and F, as follows:

E = [x_11 x_12… x_1n

x_21 x_22… x_2n

…………

x_k1 x_k2… x_kn]

F = [x_11′ x_12’… x_1n’

x_21′ x_22’… x_2n’

…………

x_k1′ x_k2’… x_kn’]

Here, k is the number of samples for each sampling.

To solve the first-order sensitivity coefficient and the global sensitivity coefficient, based on the two matrices E and F, the matrix E_r’ is constructed, where i = 1, 2,…, n. The matrix E_r’ is obtained by replacing the i-th column of the matrix E with the i-th column of the matrix F, and the rest is exactly the same as the matrix E. The matrix E_r’ can be expressed as follows:

E_r’ = [x_11 x_12… x_1i’… x_1n

x_21 x_22… x_2i’… x_2n

………………

x_k1 x_k2… x_ki’… x_kn]

A row of the sample matrix represents a complete set of input parameter samples. Therefore, each row of input parameters can solve a model output result separately. The first-order sensitivity coefficient S_i and the global sensitivity coefficient S_(i)^tot can be approximately calculated through the following formulas:

S_i = (1/k) ∑_(h=1)^k f(F)_h (f(E_r’)*k – f(E)k) / DS(i)^tot ≈ (1/2k) ∑*(k=1)^k (f(E)_h – f(E_r’)_h)^2 / D

Here, f(E)_h is the output result obtained by substituting the h-th row of the matrix E into the model; f(F)_h is the output result obtained by substituting the h-th row of the matrix F into the model; f(E_r’)_h is the output result obtained by substituting the h-th row of the matrix E_r’ into the model; D is the total variance of the calculation model.

**Spiral Bevel Gear Machine Tool Processing Model**

**Machine Tool Geometric Error Classification**

The structure of the spiral bevel gear CNC machine tool is shown in Figure 2. According to the set CNC program, the axes A, B, X, Y, and Z move together to process the spiral bevel gear in the Gleason system. The c-axis drives the cutter head to rotate and does not affect the tooth surface generation process. The coordinate transformation relationship of the machine tool is shown in Figure 3.

For the five motion axes related to the tooth surface generation, each axis has geometric errors due to manufacturing, assembly, and other factors, which directly lead to the generation of tooth surface deviations. Each axis has six geometric errors, including three linear errors and three angular errors. For the entire machine tool, 30 geometric errors need to be considered.

To facilitate analysis, the geometric error sequence is numbered as shown in Table 1. Here, ε represents the angular error, δ represents the linear error, and the error variables are distinguished by two subscripts: the first subscript represents the direction of the error, and the second subscript indicates the axis where the error is located. For example, ε_ay represents the angular error of the Y-axis in the α direction, and ε_yx represents the linear error of the X-axis in the y direction.

Axis | Geometric Error | Number | ||
---|---|---|---|---|

X | δ_xx | δ_x | δ_x | 1 – 3 |

Y | δ_yx | δ_y | δ_y | 4 – 6 |

Z | δ_zx | δ_z | δ_z | 7 – 9 |

A | ε_ax | ε_A | ε_A | 10 – 12 |

BXYZ | ε_Bx | ε_By | ε_Bz | 13 – 15 |

ε_ax | ε_Ax | ε_Byx | 16 – 18 | |

ε_aY | ε_Ay | ε_ByY | 19 – 21 | |

ε_aZ | ε_AZ | ε_ByZ | 22 – 24 | |

ε_a | ε_A | ε_By | 25 – 27 | |

B | ε_aB | ε_B | ε_ByB | 28 – 30 |

**Tooth Surface Processing Process**

The generating motion of the workpiece gear is determined by the five axes X, Y, Z, A, and B. Therefore, the homogeneous transformation matrix from the A-axis to the Y-axis combined with the tool equation can obtain the tooth surface equation r_B under ideal conditions.

r_z = M_A · M_B · M_Z · M_X · M_Y · r_t(u, θ)

Here, M (q = X, Y, Z, A, B) is the motion transformation matrix corresponding to each axis; r is the expression obtained from the tool equation; u and θ are the parameters of the cutter head. Due to space limitations, the specific process can refer to literature [15].

Considering the influence of machine tool geometric errors in actual processing, the actual tooth surface equation is as follows:

r_g^e = M_A · M_A^e · M_B · M_B^e · M_Z · M_Z^e · M_x · M_x^e · M_Y · M_Y^e · r_1(u, θ)

Here, M_q (q = X, Y, Z, A, B) is the geometric error matrix corresponding to each axis.

**Sensitivity Analysis Example**

**Sampling Calculation**

Taking the gear shown in Table 2 as an example, the theoretical tooth surface of the spiral bevel gear can be obtained according to equation (13). Considering the influence of machine tool geometric errors in actual processing, the actual tooth surface of the spiral bevel gear can be calculated according to equation (14).

Parameter | Value |
---|---|

Number of Teeth Z1 | 18 |

Large End Module m/mm | 4.29 |

Tooth Face Width/mm | 45 |

Hand of Rotation | Left-handed |

Middle Point Helix Angle/() | 35 |

Addendum h/mm | 8.78 |

Dedendum h/mm | 5.03 |

Pitch Cone Angle y/() | 27.21 |

Face Cone Angle/() | 31.65 |

Root Cone Angle/() | 25.36 |

Outer Cone Distance L/mm | 140.9 |

To facilitate analysis and calculation, the tooth surface is represented by a discrete 15X9 lattice. Combining equations (13) and (14), the theoretical tooth surface lattice and the actual tooth surface lattice can be obtained. Through the coordinate values of the corresponding points in the lattice, the deviation value K_f of the corresponding tooth surface point can be solved. K represents the tooth surface deviation, which is the output result of the calculation model and is used to measure the magnitude of the tooth surface deviation. It can be obtained through the following formula:

K = ∑_(j=1)^13 |K_j|

Given that the change amount of each linear parameter is +0.01 mm and the change amount of each angular parameter is +27″, taking the tooth surface deviation K as the model output result, the local sensitivity analysis result is shown in Figure 4.

[Insert Image of Local Sensitivity Analysis Result]

Referring to the change amount of the input parameters in the local sensitivity analysis, the linear error range is given as 0 – 10 μm, the angular error range is 0″ – 27″, and the geometric error parameters conform to a uniform distribution. Write a sampling program to randomly sample within the given range of geometric errors to generate the machine tool geometric error parameters. Combining equations (13) and (14), the corresponding theoretical tooth surface and actual tooth surface can be calculated, and according to the two tooth surface lattices, the corresponding tooth surface deviation K can be solved.

Through the Sobol global sensitivity analysis method, the global sensitivity coefficients of 30 machine tool geometric errors can be solved. Arranging them in descending order according to the global sensitivity coefficient。

**Conclusions**

Through applying the two sensitivity analysis methods to the sensitivity analysis of the machining deviation model of the spiral bevel gear tooth surface, the following conclusions can be drawn:

- In the ideal case where the value range and distribution law of the input parameters are exactly the same, the local sensitivity analysis method has a smaller calculation amount and can obtain preliminary analysis results more quickly. At the same time, the distribution law of this analysis result is the same as that of the global sensitivity analysis result, which has certain reference value.
- The Sobol global sensitivity analysis method takes into account the influence of the value range and distribution law of the input parameters on the analysis result, which is more scientific, and the obtained conclusion is more credible, and the applicable range is more extensive. In addition, most global sensitivity analyses can be used to study the quantitative relationship between the input parameters and the output result and the influence of the interaction between the parameters on the output result. However, compared with the local sensitivity analysis method, the calculation amount is significantly increased, and more time is required.
- For models with unclear distribution range of input parameters, less available time, and the same or similar distribution of linear or nonlinear models, the local sensitivity analysis can be considered; for nonlinear models with complex distribution range of input parameters and high accuracy requirements, the global sensitivity analysis method can be considered.

This research provides theoretical guidance for the use of sensitivity analysis methods in the machining of spiral bevel gear tooth surfaces and offers a reference for the selection of sensitivity analysis methods.

**Future Research Directions**

Although this study has made some progress in the sensitivity analysis of the machining deviation of spiral bevel gear tooth surfaces, there are still some areas that need further exploration and research:

- Further study the influence of other factors on the tooth surface deviation of spiral bevel gears, such as the wear of the cutting tool, the vibration of the machine tool, and the change of the processing environment.
- Explore more advanced sensitivity analysis methods and techniques to improve the accuracy and efficiency of the analysis.
- Combine the sensitivity analysis results with the optimization design of the machine tool to further improve the machining accuracy and quality of the spiral bevel gear.
- Conduct more practical experiments and verifications to validate the reliability and practicability of the sensitivity analysis results.

In conclusion, the sensitivity analysis of the machining deviation of spiral bevel gear tooth surfaces is a complex and important research topic. Further research in this area will help improve the manufacturing accuracy and performance of spiral bevel gears, and promote the development of related industries.