Helicopter multi-configuration transmission chain optimization method adopting response surface method
By analyzing the sensitivity of helicopter transmission chain gear parameters using response surface methodology, an optimization model was constructed, which solved the problems of long design cycles and low efficiency in traditional methods, and achieved efficient transmission chain optimization and design guidance.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-04
- Publication Date
- 2026-03-13
AI Technical Summary
Traditional helicopter transmission chain optimization methods rely on engineering experience and costly simulations, making it difficult to efficiently obtain the global optimal solution and to optimize under multiple constraints.
The response surface methodology was used to construct regression equations to analyze the sensitivity of gear parameters to target performance, key parameters were screened, an optimization model was constructed, and the optimal solution was found through optimization algorithms.
Significantly shorten the design cycle, improve design efficiency and effectiveness, ensure the feasibility and relevance of optimization results, and provide effective guidance for transmission system design.
Smart Images
Figure CN121659525A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of helicopter drivetrain optimization, and in particular to a helicopter multi-configuration drivetrain optimization method using response surface methodology. Background Technology
[0002] The helicopter's main drivetrain is a core component of the power system. Its weight directly affects the helicopter's payload, fuel economy, and overall performance, while its efficiency directly impacts core indicators such as rotor power, fuel economy, and range. Drivetrains typically include various gear configurations (such as parallel shafts and planetary gear systems), and their design involves multiple design variables (such as module, face width factor, number of teeth, and helix angle), while also meeting multiple constraints related to strength, lifespan, and NVH (noise, vibration, and harshness). Traditional optimization methods often rely on engineers' experience and trial-and-error parameter adjustments, or employ computationally expensive finite element analysis and multibody dynamics simulations for iterative processing. These methods are cumbersome, time-consuming, and struggle to comprehensively assess the impact of each parameter on system weight and efficiency, making it difficult to efficiently obtain globally optimal or near-optimal solutions under multiple constraints. Summary of the Invention
[0003] To address the above problems, this invention proposes a helicopter multi-configuration transmission chain optimization method using response surface methodology. The method uses response surface methodology to fit regression equations to analyze the sensitivity of helicopter main transmission chain gear parameters to target performance. Based on the sensitivity, the gear parameters to be optimized are selected, and the regression equations are fitted using response surface methodology as the target equations for optimization.
[0004] The technical solution of this invention is as follows: It is carried out according to the following steps:
[0005] Step 1: Determine the optimization objective and design variables;
[0006] Define the configuration of the helicopter's main drivetrain and identify one or more key performance indicators as optimization targets; select key gear parameters as design variables based on their impact on the target performance.
[0007] The design variables include, but are not limited to, gear module, tooth width (or tooth width coefficient), number of teeth, helix angle, and pressure angle; the target performance indicators include, but are not limited to, weight and efficiency.
[0008] Step 2: Define the upper and lower limits of variables, experimental design, and sample point generation;
[0009] For the design variables determined in step 1, based on engineering design experience, process feasibility, and installation space constraints, the value range of each design variable is determined, i.e., its upper and lower limits are defined. On this basis, experimental design methods are used to generate a series of representative sample points within this constrained design space; the experimental design methods include, but are not limited to, central composite design, Box-Behnken design, or Latin hypercube design.
[0010] Step 3: Construct a target response surface model for sensitivity analysis;
[0011] For each sample point generated in step 2, one or more target performance indicators are calculated based on system design theory and target performance calculation model.
[0012] Using the response surface methodology, with the design variables as input and the selected target performance index as output, a set of explicit and approximate functional relationships are fitted to obtain the first response surface regression equation. By analyzing the regression coefficients and significance levels of the design variables in each regression equation or by performing analysis of variance, the sensitivity of each design variable to different target performance indexes is quantitatively assessed, and key influencing variables are identified.
[0013] Step 4: Screening of Key Design Variables
[0014] Based on the sensitivity analysis results obtained in step 3, several design variables that have the most significant impact on the target performance of the transmission chain are selected as key optimization variables in the subsequent optimization process.
[0015] Step 5: Construct an optimized target response surface model:
[0016] Using the key optimization variables selected in step 4 as input, the experimental design method is used again to generate a second batch of sample points in its design space; for each new sample point, the other unselected parameters remain unchanged, and its target performance of the transmission chain is calculated; using the response surface method, the second response surface regression equation with the key optimization variables as input and the target performance of the transmission chain as output is fitted, and this equation is used as the objective function of the optimization process;
[0017] Step 6: Define constraints:
[0018] Based on the design requirements of the helicopter transmission system, the constraints of the optimization problem are defined; the constraints include, but are not limited to, contact strength constraints, bending strength constraints, overlap ratio constraints, transmission ratio constraints, structural dimension boundary constraints, and maximum linear velocity of the gears.
[0019] Step 7: Perform optimization solution:
[0020] Using the second response surface regression equation constructed in step 5 as the objective function and the requirements defined in step 6 as constraints, a complete optimization mathematical model is constructed; the optimization algorithm is used to solve the mathematical model to find the result that optimizes the target performance of the transmission chain; the optimization algorithm includes sequential quadratic programming, genetic algorithm or feasible direction method.
[0021] Step 3 specifically involves:
[0022] Assuming the response equation of each parameter to the target performance is G(X), the approximate response equation obtained using the response surface methodology is: Then the various parameters can be obtained. Structural response equation Sensitive factors for:
[0023]
[0024] Then a certain design parameter Sensitivity percentage for:
[0025]
[0026] According to the above formula, the sensitivity percentage is used as an indicator to evaluate the sensitivity of each parameter to the target performance.
[0027] As an alternative to step 3, after completing the experimental design and sample point generation, the absolute value of the Pearson correlation coefficient (r) between each parameter of the main drivetrain and the target performance can also be used as a comparison standard to quantify the strength of the correlation between the two, which is a method of sensitivity analysis. The formula is as follows:
[0028]
[0029] Then a certain design parameter X k The percentage of the correlation coefficient ρ Xk (k∈i) is:
[0030] .
[0031] Step 5: When multiple optimization objectives are selected, each objective is multiplied by a set of weighting coefficients according to its different importance, and then the sums are used as the objective function. The optimal solution is then calculated from this sum. Its mathematical expression is:
[0032]
[0033] in, These are called weighting coefficients.
[0034] Compared with the prior art, the advantages of the present invention are as follows:
[0035] By constructing a low-computational-cost response surface surrogate model to replace complex physical simulations or high-precision computational models, the number of analyses required in the optimization process is significantly reduced, shortening the design cycle. For the first time, the sensitivity of each gear parameter to the target performance is systematically analyzed using the response surface methodology, helping designers identify key factors affecting the target performance, making optimization more targeted and avoiding blind optimization. This method can handle complex transmission chain systems with multiple gear configurations, describing the relationship between target performance and design variables through explicit mathematical equations, facilitating the integration of multiple constraints and efficient optimization. The final optimization results are validated using a practical model, significantly improving the efficiency and effectiveness of complex transmission system design, ensuring the feasibility of the solution, and providing direct and effective guidance for the design of helicopter transmission systems. Attached Figure Description
[0036] Figure 1 This is a flowchart of a method for optimizing the transmission chain of helicopters with multiple configurations.
[0037] Figure 2 This is a structural schematic diagram of an embodiment.
[0038] Figure 3 This is a Pareto chart plotted as a percentage of the correlation coefficients between each parameter and the efficiency in the embodiment. Detailed Implementation
[0039] To clearly illustrate the technical features of the present invention, the present invention will be described in detail below through specific embodiments and in conjunction with the accompanying drawings.
[0040] Step 1: Determine the optimization objective and design variables;
[0041] Define the configuration of the helicopter's main drivetrain and identify one or more key performance indicators as optimization targets; select key gear parameters as design variables based on their impact on the target performance.
[0042] The design variables include, but are not limited to, gear module, tooth width (or tooth width coefficient), number of teeth, helix angle, and pressure angle.
[0043] Target performance metrics include, but are not limited to, weight and efficiency.
[0044] Step 2: Define the upper and lower limits of variables, experimental design, and sample point generation;
[0045] For the design variables determined in step 1, based on engineering design experience, process feasibility, and installation space constraints, the value range of each design variable is determined, i.e., its upper and lower limits are defined. On this basis, experimental design methods are used to generate a series of representative sample points within this constrained design space; the experimental design methods include, but are not limited to, central composite design, Box-Behnken design, or Latin hypercube design.
[0046] (1) Central composite experimental design
[0047] Central composite design is suitable for situations where the output response has a nonlinear relationship with the parameter to be corrected. A complete central composite design requires n=2 tests. k +2k+n c In this experiment, k, the number of factors, needs to be determined based on the number of parameters to be corrected. k This indicates the number of trials in a 2-level full factorial design, where 2k represents the number of pivot points, and n... c This indicates the number of center points in a central composite experiment. As the value of k increases, the number of points n increases rapidly, so the number of k should not be too large. When the number of k is large, a partial central composite experimental design can be used. Taking a 3-factor, 1-center design as an example: Cube points: all points have coordinates of -1 or 1, totaling 8; Center points: all points have coordinates of 0 in each dimension, which can be added as needed; Axis points: except for one independent variable whose coordinate is 'a', all other coordinates are 0, totaling 2k = 6.
[0048] (2) BBD experimental design
[0049] BBD experimental design is suitable for situations where the output response has a nonlinear relationship with the parameter to be corrected. When the level range of the experimental factor is limited, making it impossible to conduct experiments at axial points, BBD experimental design can be used. In BBD experiments, the test points are distributed at the center points of each edge of the cube, and axial point experiments are not required. Therefore, this method is suitable for experimental designs with limited experimental factor levels. BBD experiments can perform statistical analysis on experimental data and provide graphical analysis of continuous characteristics, allowing for a simple and clear understanding of the correspondence between factors and responses. In a 3-factor, 1-center-point experimental design, the test points are distributed at the center points of each edge of the cube, totaling 12 points; there is also one center point.
[0050] (3) Latin hypercube design
[0051] It employs the principle of equally probable random orthogonal distribution, enabling the acquisition of highly accurate response surface equations with a minimal number of experimental points. When using the Latin hypercube design, the number of experimental points can be arbitrarily specified, as long as it is not less than the minimum number of experimental points required to determine the unknown parameters; designs with any number of experimental points can be generated using this method. For the design variable x = (x1, ..., x...) n ,)T To obtain m sample points, the domain of each design variable can be divided into m non-overlapping intervals according to the principle of equal probability. A value of x can be randomly selected from each interval to obtain n sets of data. = (x i 1 , ..., x i m ) T (i=1, ..., n). By randomly combining the elements in these n data sets without repetition, we can obtain m sample points.
[0052] Step 3: Construct the target response surface model (sensitivity analysis);
[0053] For each sample point generated in step 2, one or more target performance indicators are calculated based on system design theory and target performance calculation model.
[0054] Using the response surface methodology, with the design variables as input and the selected target performance index as output, a set of explicit and approximate functional relationships are fitted to obtain the first response surface regression equation. By analyzing the regression coefficients and significance levels of the design variables in each regression equation or by performing analysis of variance, the sensitivity of each design variable to different target performance indexes is quantitatively assessed, and key influencing variables are identified.
[0055] Assuming the response equation of each parameter to the target performance is G(X), the approximate response equation obtained using the response surface methodology is: Then the various parameters can be obtained. Structural response equation Sensitive factors for:
[0056]
[0057] Then a certain design parameter Sensitivity percentage for:
[0058]
[0059] According to the above formula, the sensitivity percentage is used as an indicator to evaluate the sensitivity of each parameter to the target performance.
[0060] As an alternative to step S3, after completing the experimental design and sample point generation, the absolute value of the Pearson correlation coefficient (r) between each parameter of the main drivetrain and the target performance can also be used as a comparison standard to quantify the strength of the correlation between the two, which is a method of sensitivity analysis. The formula is as follows:
[0061]
[0062] Then a certain design parameter X k The percentage of the correlation coefficient ρ Xk (k∈i) is:
[0063] .
[0064] Step 4: Screening of Key Design Variables
[0065] Based on the sensitivity analysis results obtained in step 3, several design variables that have the most significant impact on the target performance of the transmission chain are selected as key optimization variables in the subsequent optimization process.
[0066] Step 5: Construct an optimized target response surface model:
[0067] Using the key optimization variables selected in step 4 as input, a second batch of sample points is generated within its design space using the experimental design method. For each new sample point, other unselected parameters remain unchanged, and its target performance of the transmission chain is calculated. Using the response surface methodology, a second response surface regression equation is fitted with the key optimization variables as input and the target performance of the transmission chain as output, and this equation is used as the objective function of the optimization process.
[0068] When optimizing, if multiple optimization objectives are selected, the fitting is the relationship between the weighted results of different objectives and the parameters. When there is only one objective, the fitting is the relationship between the single objective and the parameters.
[0069] When multiple optimization objectives are selected, each objective is multiplied by a set of weighting coefficients according to its different importance, and then the sums are used as the objective function to find the optimal solution. Its mathematical expression is:
[0070]
[0071] in, These are called weighting coefficients.
[0072] Step 6: Define constraints:
[0073] Based on the design requirements of the helicopter transmission system, the constraints of the optimization problem are defined; the constraints include, but are not limited to, contact strength constraints, bending strength constraints, overlap constraints, transmission ratio constraints, structural dimension boundary constraints, and maximum linear velocity of the gears.
[0074] Step 7: Perform optimization solution:
[0075] Using the second response surface regression equation constructed in step 5 as the objective function and the requirements defined in step 6 as constraints, a complete optimization mathematical model is constructed; the optimization algorithm is used to solve the mathematical model to find the result that optimizes the target performance of the transmission chain; the optimization algorithm includes sequential quadratic programming, genetic algorithm or feasible direction method.
[0076] Step 8: Design Validation:
[0077] Substitute the optimal values of the key design variables obtained in step 7 into the original gear design and target performance calculation model for verification calculation to ensure that the optimization results meet all constraints and that the target performance is indeed optimized.
[0078] Example:
[0079] like Figure 2 This is a schematic diagram of the main reduction gear transmission of a helicopter, which uses a first-stage bevel gear set, a second-stage helical gear set, and a third-stage herringbone gear set. The first-stage bevel gear is a coaxial multi-gear ordinary transmission, the second-stage helical gear is a torque-split transmission, and the third-stage herringbone gear set is a parallel transmission.
[0080] Obtain the operating parameters of the helicopter: input power is 16000kW, rotor shaft power is 14710kW, and input speed is 8300r / min. Obtain the gear parameters of each stage of the helicopter's transmission chain: the first-stage bevel gear has a module of 6, a driving gear with 28 teeth, a face width coefficient of 0.32, a transmission ratio of 3.21, a helix angle of 40°, and a pressure angle of 25°; the second-stage helical gear has a module of 4.5, a driving gear with 32 teeth, a face width coefficient of 0.31, a transmission ratio of 3.47, a helix angle of 35°, and a pressure angle of 25°; the third-stage bevel gear has a module of 6.2, a driving gear with 23 teeth, a face width coefficient of 0.75, a transmission ratio of 5.6, a helix angle of 30°, and a pressure angle of 25°.
[0081] In a multi-objective optimization problem involving weight and efficiency, the weight M and efficiency η are dimensionless. Since efficiency is usually maximized, normalization transforms the problem into minimization. The comprehensive objective function is constructed as follows:
[0082]
[0083] In the formula, w is the weighting coefficient for weight; M ref The initial calculated weight, in kg; ƞ ref This is the weight obtained from the initial calculation.
[0084] Based on the calculation results, a model of the overall configuration of the main drive train of the transmission system is fitted, and the sensitivity of the parameters of each gear stage to the normalized weight and efficiency of the main drive train under the set weights is analyzed.
[0085] The module, number of teeth, face width factor, transmission ratio, helix angle, and pressure angle were used as variables for normalized weight and efficiency estimation in the experimental design. To more accurately analyze the sensitive parameters of the main drivetrain, eighteen variables were subjected to 2... 18 The experimental design for this run is used to fit the first response surface regression equation as follows, in order to perform transmission chain sensitivity parameter analysis.
[0086]
[0087] Figure 3 The results are based on sensitivity analysis using regression equations.
[0088] Step 5: Select the design parameters as follows based on design requirements: module and number of teeth for the first-stage bevel gear, module and number of teeth for the second-stage cylindrical gear, and module and number of teeth for the third-stage cylindrical gear. For the optimization of transmission chain sensitive parameters based on the response surface methodology, the Box-Behnken design experiment is used. A second-order polynomial is used to fit the function, removing polynomials with coefficients of 0 and polynomials with excessively large p-values. The fitted second response surface regression equation is as follows:
[0089]
[0090] The regression analysis results are shown in Table 2.16.
[0091] Table 2.16 Regression Analysis
[0092] Estimate SE tStat pValue <![CDATA[m1 2 ]]> 0.0271 0.0086 3.1647 0.0030 <![CDATA[z1 2 ]]> 0.0014 0.0003 4.1034 0.0002 <![CDATA[m2 2 ]]> 0.0117 0.0086 1.3653 0.1800 <![CDATA[z2 2 ]]> 0.0009 0.0003 2.6643 0.0112 <![CDATA[z3 2 ]]> 0.0006 0.0003 1.7225 0.0929 <![CDATA[m1z1]]> 0.0165 0.0020 8.3789 0.0000 <![CDATA[m2z2]]> 0.0102 0.0020 5.2028 0.0000 <![CDATA[m3z3]]> 0.0101 0.0020 5.1314 0.0000 <![CDATA[m1]]> -0.5986 0.1076 -5.5647 0.0000 <![CDATA[z1]]> -0.1353 0.0234 -5.7700 0.0000 <![CDATA[m2]]> -0.2873 0.1076 -2.6712 0.0110 <![CDATA[z2]]> -0.0856 0.0241 -3.5571 0.0010 <![CDATA[m3]]> -0.1592 0.0650 -2.4499 0.0189 <![CDATA[z3]]> -0.0645 0.0241 -2.6814 0.0107 1 4.6856 0.8765 5.3456 0.0000
[0093] The constraints are as follows:
[0094] Transmission ratio constraint:
[0095]
[0096] In the formula, n in Input engine speed, n xy This refers to the rotor speed of the helicopter.
[0097] The contact strength constraint is:
[0098]
[0099] In the formula, For tooth contact stress, N / mm 2 ; The contact limit stress is N / mm. 2 S Hmin This is the minimum safety factor for contact strength.
[0100] The bending strength constraint is:
[0101]
[0102] In the formula, To calculate the calculated bending stress of the gear, N / mm 2 ; The bending limit stress is N / mm. 2 ; This is the minimum safety factor for bending strength.
[0103] Bond strength constraints:
[0104]
[0105] In the formula, The integral temperature of the gear is expressed in °C. This refers to the gear bonding temperature, expressed in °C. This represents the minimum safety factor.
[0106] Maximum diameter constraint:
[0107]
[0108]
[0109] In the formula, m is the gear module (mm); z is the number of gear teeth; d max The value is the maximum gear diameter, in mm.
[0110] Maximum linear velocity constraint:
[0111]
[0112]
[0113] In the formula, n is the gear speed, r / min; v max The maximum gear linear velocity is in mm.
[0114] For the sensitive parameters of the transmission chain in the cylindrical gear torque splitting transmission configuration, a sequential quadratic programming algorithm was adopted. After optimization, the module of the first-stage bevel gear is 2.68 and the number of teeth is 44, the module of the second-stage cylindrical gear is 4.10 and the number of teeth is 32, and the module of the third-stage cylindrical gear is 1.5 and the number of teeth is 45. Before optimization, the weight of the main transmission chain gears in the transmission system was 620.8 kg, and the efficiency of the main transmission chain was 98.78%. After optimization, the gear weight is 287.6 kg, a weight reduction of 53.67%, and the efficiency of the main transmission chain is 99.06%, an efficiency improvement of 0.276%.
[0115] There are many specific ways to implement this invention. The above description is only a preferred embodiment of this invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of this invention, and these improvements should also be considered within the scope of protection of this invention.
Claims
1. A helicopter multi-configuration transmission train optimization method using response surface methodology, characterized in that, Follow these steps: Step 1: Determine the optimization objective and design variables; Define the configuration of the helicopter's main drivetrain and identify one or more key performance indicators as optimization targets; Based on their impact on the target performance, key gear parameters are selected as design variables; The design variables include, but are not limited to, gear module, tooth width (or tooth width coefficient), number of teeth, helix angle, and pressure angle; the target performance indicators include, but are not limited to, weight and efficiency. Step 2: Define the upper and lower limits of variables, experimental design, and sample point generation; For the design variables determined in step 1, based on engineering design experience, process feasibility, and installation space constraints, the value range of each design variable is determined, i.e., its upper and lower limits are defined. On this basis, experimental design methods are used to generate a series of representative sample points within this constrained design space; the experimental design methods include, but are not limited to, central composite design, Box-Behnken design, or Latin hypercube design. Step 3: Construct a target response surface model for sensitivity analysis; For each sample point generated in step 2, one or more target performance indicators are calculated based on system design theory and target performance calculation model. Using the response surface methodology, with the design variables as input and the selected target performance index as output, a set of explicit and approximate functional relationships are obtained, namely the first response surface regression equation. By analyzing the regression coefficients and significance levels of the design variables in each regression equation, or by performing analysis of variance, the sensitivity of each design variable to different target performance indicators can be quantitatively assessed, and key influencing variables can be identified. Step 4: Screening of key design variables: Based on the sensitivity analysis results obtained in step 3, several design variables that have the most significant impact on the target performance of the transmission chain are selected as key optimization variables in the subsequent optimization process. Step 5: Construct an optimized target response surface model: Using the key optimization variables selected in step 4 as input, the experimental design method is used again to generate a second batch of sample points in its design space; for each new sample point, the other unselected parameters remain unchanged, and its target performance of the transmission chain is calculated; using the response surface method, the second response surface regression equation with the key optimization variables as input and the target performance of the transmission chain as output is fitted, and this equation is used as the objective function of the optimization process; Step 6: Define constraints: Based on the design requirements of the helicopter transmission system, the constraints of the optimization problem are defined; the constraints include, but are not limited to, contact strength constraints, bending strength constraints, overlap ratio constraints, transmission ratio constraints, structural dimension boundary constraints, and maximum linear velocity of the gears. Step 7: Perform optimization solution: Using the second response surface regression equation constructed in step 5 as the objective function and the requirements defined in step 6 as constraints, a complete optimization mathematical model is constructed; the optimization algorithm is used to solve the mathematical model to find the result that optimizes the target performance of the transmission chain; the optimization algorithm includes sequential quadratic programming, genetic algorithm or feasible direction method.
2. The helicopter multi-configuration drivetrain optimization method using response surface methodology as described in claim 1, characterized in that, Step 3 specifically involves: Assuming the response equation of each parameter to the target performance is G(X), the approximate response equation obtained using the response surface methodology is: Then the various parameters can be obtained. Structural response equation Sensitive factors for: ； Then a certain design parameter Sensitivity percentage for: ； According to the above formula, the sensitivity percentage is used as an indicator to evaluate the sensitivity of each parameter to the target performance.
3. The helicopter multi-configuration drivetrain optimization method using response surface methodology according to claim 1, characterized in that, As an alternative to step 3, after completing the experimental design and sample point generation, the absolute value of the Pearson correlation coefficient (r) between each parameter of the main drivetrain and the target performance can also be used as a comparison standard to quantify the strength of the correlation between the two, which is a method of sensitivity analysis. The formula is as follows: ； Then a certain design parameter X k The percentage of the correlation coefficient ρ Xk (k∈i) is: 。 4. The helicopter multi-configuration drivetrain optimization method using response surface methodology as described in claim 1, characterized in that, Step 5, when multiple optimization objectives are selected, includes, but is not limited to, multiplying each objective by a set of weighting coefficients according to its different importance, then summing them as the objective function, and then finding the optimal solution. The mathematical expression for this is: ； in, These are called weighting coefficients.