Nonlinear rotor system dynamic response and sensitivity calculation method, device, equipment and medium

Abstract

This application discloses a method, apparatus, equipment, and medium for calculating the dynamic response and sensitivity of a nonlinear rotor system. The method includes the following steps: S1, based on the structural characteristics of the nonlinear rotor system, performing dynamic modeling of the nonlinear rotor system and establishing the equations of motion for the nonlinear rotor system; S2, establishing the hypercomplex form of the equations of motion for the nonlinear rotor system to obtain the hypercomplex equations of motion for the nonlinear rotor system, using complex matrix expression, and expanding the dimension of the complex matrix equations of motion for the nonlinear rotor system to convert them into equivalent real matrix equations; S3, solving the established hypercomplex equations of motion for the nonlinear rotor system, extracting the real and imaginary parts of the calculation results, and obtaining the system response and corresponding sensitivities of each order. This application achieves automatic updating and solving of the Jacobian matrix during the iteration process, ensuring the accuracy of the calculation results, and can simultaneously calculate the response and dynamic response sensitivity of the nonlinear rotor system, resulting in high computational efficiency.

Description

Technical Field

[0001] This application relates to the field of aero-engine technology, and in particular to methods, apparatus, equipment and media for calculating the dynamic response and sensitivity of nonlinear rotor systems. Background Technology

[0002] Currently, the dynamic response sensitivity of aero-engine rotors is essential for designers to quantitatively understand the impact of design parameters on rotor vibration. It also plays a crucial role in rotor system model correction, parameter identification, and dynamic optimization. The complex contact dynamics between the rotor and its supports lead to significant nonlinear phenomena in the system, and these nonlinear contact parameters have a significant impact on the rotor system's motion state and stability. Therefore, a stable, accurate, and efficient method for calculating nonlinear dynamic response sensitivity is crucial.

[0003] Existing technologies mainly include three types of dynamic response sensitivity calculation methods, namely:

[0004] Direct Differential Method: Directly differentiate the differential equation of rotor structure motion with respect to the design parameters to derive the sensitivity differential equation.

[0005] Finite difference method: Introduce small perturbations to the design variables and use the difference quotient to approximate the sensitivity.

[0006] Complex variable differentiation method: Add a small imaginary perturbation to the design variable, and obtain the sensitivity calculation result by extracting the imaginary part of the output.

[0007] However, existing methods for calculating dynamic response sensitivity have the following problems:

[0008] The traditional direct differentiation method is the most direct method. However, when using the direct differentiation method to calculate the sensitivity of the dynamic response of a system to different parameters or different orders of the same parameter, it is necessary to derive different differential equations. At the same time, there are many nonlinear parameters that affect the dynamic response in a rotor system, and these parameters usually exhibit multiple nonlinear forms. This makes the process of calculating the dynamic response sensitivity of a rotor system using the direct differentiation method redundant, complex, and prone to errors, which greatly reduces the computational efficiency.

[0009] The sensitivity calculation results using the finite difference method are very unstable under different perturbation step sizes. Therefore, this method will cause a lot of time cost due to the uncertainty of the perturbation step size of the design variables and the multiple calculations of the dynamic equations, resulting in poor computational accuracy and efficiency.

[0010] The complex variable differentiation method cannot accurately calculate higher-order sensitivity and mixed sensitivity of multiple parameters. Therefore, when performing higher-order sensitivity analysis of parameters or mixed sensitivity analysis involving multiple parameters, the traditional complex variable differentiation method may not yield accurate results.

[0011] Meanwhile, when solving the motion differential equations of a nonlinear rotor system, it is generally used Newmark-β The method is similar to the Newton-Raphson method. However, when the nonlinear form is complex, it is very difficult to accurately solve the Jacobian matrix during the iteration process, which will result in inaccurate calculation results of subsequent response and dynamic response sensitivity. Summary of the Invention

[0012] This application provides a method for calculating the dynamic response and sensitivity of nonlinear rotor systems, which solves the technical problems of low calculation efficiency, insufficient calculation accuracy, and inaccurate results in existing dynamic response and sensitivity calculation methods.

[0013] This application is achieved through the following solution:

[0014] A method for calculating the dynamic response and sensitivity of a nonlinear rotor system, including the following steps:

[0015] S1. Based on the structural characteristics of the nonlinear rotor system, perform dynamic modeling of the nonlinear rotor system and establish the motion equations of the nonlinear rotor system;

[0016] S2. Establish the hypercomplex form of the motion equations of the nonlinear rotor system to obtain the hypercomplex motion equations of the nonlinear rotor system. Use complex matrix expression to expand the dimension of the motion equations of the nonlinear rotor system of complex matrix into an equivalent real matrix.

[0017] S3. Solve the hypercomplex equations of motion of the established nonlinear rotor system, extract the real and imaginary parts of the calculation results, and obtain the response of the nonlinear rotor system and the corresponding sensitivities of each order or mixed sensitivities.

[0018] Furthermore, in step S1, based on the structural characteristics of the nonlinear rotor system, when performing dynamic modeling of the nonlinear rotor system, the system's mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear force vector F are obtained. nl Resultant external force vector F u Establish the equations of motion for the nonlinear rotor system:

[0019] ;

[0020] in, , and Let these represent the system's acceleration, velocity, and displacement vectors, respectively. This indicates the system rotational speed.

[0021] Furthermore, step S2 specifically includes the following steps:

[0022] S21. Add imaginary perturbations to the response and design parameters respectively, where the added perturbations are along different imaginary directions, and the unit of the imaginary direction is . , We obtain the hypercomplex form of the system's mass matrix M, stiffness matrix K, damping matrix C, and gyroscope matrix G;

[0023] S22. Calculate the nonlinear force using the response and parameters after adding the perturbation, and obtain the nonlinear restoring force F. nl hypercomplex form;

[0024] S23. Combine the mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear restoring force vector F. nl Resultant external force vector F u Substituting the hypercomplex form into the equations of motion of the nonlinear rotor system, we obtain the hypercomplex form of the equations of motion of the nonlinear rotor system:

[0025] ;

[0026] in, , , , , and These are the hypercomplex forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, nonlinear force vector, and resultant external force vector, respectively, after adding imaginary perturbations along the imaginary direction;

[0027] S24. The parameters in the obtained hypercomplex form of the nonlinear rotor system's motion equations are equivalently represented in the real-field matrix form to obtain the dimension-expanded equivalent form of the motion equations:

[0028] ;

[0029] in , , , , and These are the equivalent matrix forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, eccentric force vector, and nonlinear force vector of the multi-complex domain after perturbation.

[0030] Furthermore, step S3 specifically includes the following steps:

[0031] S31. Determine the residual R of the equation, and use R... * To express the hypercomplex form of this residual, using Newmark-β The method is used to solve the equation of motion, and then x is... * n The value of x* n+1 The calculation yields the following formula, which determines the residual R of the equation of motion. * :

[0032] ;

[0033] In the formula, Let be the hypercomplex form of the equivalent resultant external force vector at the next moment. The equivalent stiffness is in hypercomplex form:

[0034] ;

[0035] ;

[0036] In the formula, for Newmark-β The integration constants are typically taken as 0.25 and 0.5.

[0037] S32. Determine the Jacobian matrix of the equation using J. * Let J be the hypercomplex form of this Jacobian matrix. The Jacobian matrix J can be determined according to the following formula. * :

[0038] ;

[0039] In the formula, Indicates the response x at the th k The residuals of the equations after adding imaginary perturbations to each degree of freedom express The k One degree of freedom along The coefficient in the imaginary part direction;

[0040] Using complex matrix representation, the Jacobian matrix is ​​expanded to an equivalent real matrix. The Jacobian matrix after dimension expansion with added imaginary perturbation is represented as follows:

[0041] ;

[0042] S33, Utilizing Equation Residuals And Jacobi matrix The equations are calculated to obtain the Jacobian matrix. after, The remaining value is represented as Using a complex matrix representation, this matrix is ​​expanded to an equivalent real matrix as shown below. This represents the real matrix after dimension expansion of the new residual vector following the addition of the imaginary part perturbation, as shown below:

[0043] ;

[0044] S34. Repeat the above steps to perform Newton's iterative calculation of the equations of motion according to the following formula until the hypercomplex form of the solution to the equations of motion is obtained:

[0045] ;

[0046] in, This is the response value from the previous moment. Let Jacobian matrix be the value from the previous time step. The remaining value of the Jacobian matrix calculated using the residual R at the previous time step;

[0047] S35. Extract the values ​​of the real and imaginary coefficients of the calculated response x to obtain the response of the nonlinear rotor system and the corresponding sensitivity or mixed sensitivity of each order.

[0048] Furthermore, step S35 specifically includes the following steps:

[0049] S351. Extract the calculated response x j+1 Real part:

[0050] x=Re(x j+1 );

[0051] S352, Extracting response x j+1 Imaginary part coefficients combined with response x j+1 The real part is used to obtain the response and first- and second-order sensitivity results of the rotor system:

[0052] ;

[0053] ;

[0054] in, Indicates the dynamic response for design parameters First-order sensitivity, Indicates the dynamic response for design parameters Second-order sensitivity, Indicates the dynamic response x j+1 Along the virtual part The coefficient of direction, Indicates the dynamic response x j+1 Along the imaginary part of the mixture The coefficient of direction.

[0055] This application also provides a device for calculating the dynamic response and sensitivity of a nonlinear rotor system, including:

[0056] The equation of motion module is used to perform dynamic modeling of the nonlinear rotor system based on its structural characteristics and to establish the equation of motion for the nonlinear rotor system.

[0057] The module for solving the extended dimension of the equations of motion is used to establish the hypercomplex form of the equations of motion of a nonlinear rotor system. It obtains the hypercomplex equations of motion of the nonlinear rotor system using complex matrix expression, and expands the dimension of the complex matrix equations of motion of the nonlinear rotor system into an equivalent real matrix.

[0058] The response and sensitivity solution module is used to solve the hypercomplex motion equations of the established nonlinear rotor system, extract the real and imaginary parts of the calculation results, and obtain the response of the nonlinear rotor system and the corresponding sensitivity of each order or mixed sensitivity.

[0059] Furthermore, the module for solving the extended dimension of the equations of motion specifically includes:

[0060] The first hypercomplex form determination module is used to add imaginary perturbations to the response and design parameters, respectively. The added perturbations are along different imaginary directions, with units of . , We obtain the hypercomplex form of the system's mass matrix M, stiffness matrix K, damping matrix C, and gyroscope matrix G;

[0061] The second hypercomplex form determination module is used to calculate the nonlinear force using the response and parameters after adding the perturbation, and obtain the nonlinear restoring force F. nl hypercomplex form;

[0062] The module for determining the hypercomplex form of the equations of motion is used to determine the mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear restoring force vector F. nl Resultant external force vector F u Substituting the hypercomplex form of the equations of motion into the equations of motion of the nonlinear rotor system, we obtain the hypercomplex form of the equations of motion of the nonlinear rotor system:

[0063] ;

[0064] in, , , , , and These are the hypercomplex forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, nonlinear force vector, and resultant external force vector, respectively, after adding imaginary perturbations along the imaginary direction;

[0065] The equivalence matrix determination module is used to equivalence each parameter in the obtained hypercomplex form of the nonlinear rotor system's motion equations using matrix form in the real field, thereby obtaining the dimension-expanded equivalent form of the motion equations:

[0066] ;

[0067] in , , , , and These are the equivalent matrix forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, eccentric force vector, and nonlinear force vector of the multi-complex domain after perturbation.

[0068] Furthermore, the response and sensitivity solving module specifically includes:

[0069] The residual determination module for the equations of motion is used to determine the residual R of the equations, using R... * To express the hypercomplex form of this residual, using Newmark-β The method is used to solve the equation of motion, and then x is... * n The value of x * n+1 The calculation yields the following formula, which determines the residual R of the equation of motion. * :

[0070] ;

[0071] In the formula, Let be the hypercomplex form of the equivalent resultant external force vector at the next moment. The equivalent stiffness is in hypercomplex form:

[0072] ;

[0073] ;

[0074] In the formula, for Newmark-β The integration constants are typically taken as 0.25 and 0.5.

[0075] The Jacobian matrix determination module is used to determine the Jacobian matrix of the equation, using J... * Let J be the hypercomplex form of this Jacobian matrix. The Jacobian matrix J can be determined according to the following formula. * :

[0076] ;

[0077] In the formula, Indicates the response x at the th k The residuals of the equations after adding imaginary perturbations to each degree of freedom express The k One degree of freedom along The coefficient in the imaginary part direction;

[0078] Using complex matrix representation, the Jacobian matrix is ​​expanded to an equivalent real matrix. The Jacobian matrix after dimension expansion with added imaginary perturbation is represented as follows:

[0079] ;

[0080] The residual update module is used to utilize the equation residuals. And Jacobi matrix The equations are calculated to obtain the Jacobian matrix. after, The remaining value is represented as Using a complex matrix representation, this matrix is ​​expanded to an equivalent real matrix as shown below. This represents the real matrix after dimension expansion of the new residual vector following the addition of the imaginary part perturbation, as shown below:

[0081] ;

[0082] The module for determining the hypercomplex form of the response is used to repeat the aforementioned steps, performing Newton's iterative calculations of the equations of motion according to the following formula, until the hypercomplex form of the solution to the equations of motion is obtained:

[0083] ;

[0084] in, This is the response value from the previous moment. Let Jacobian matrix be the value from the previous time step. The remaining value of the Jacobian matrix calculated using the residual R at the previous time step;

[0085] The response and sensitivity calculation module is used to extract the values ​​of the real and imaginary coefficients of the calculated response x, and obtain the response of the nonlinear rotor system and the corresponding sensitivity of each order or mixed sensitivity.

[0086] This application also provides an electronic device, which includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method for calculating the dynamic response and sensitivity of the nonlinear rotor system.

[0087] This application also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method for calculating the dynamic response and sensitivity of the nonlinear rotor system.

[0088] This application also provides a computer program product, including a computer program or computer-executable instructions, which, when executed by a processor, implement the method for calculating the dynamic response and sensitivity of the nonlinear rotor system.

[0089] Compared with the prior art, this application can produce the following beneficial effects:

[0090] This application addresses the problem of solving for the response and dynamic response sensitivity of nonlinear rotor systems, providing a method that enables simultaneous calculation of response and sensitivity. During the calculation process, this application automatically updates and solves the Jacobian matrix, ensuring the accuracy of subsequent response and dynamic response sensitivity calculations. This invention can simultaneously calculate the response and dynamic response sensitivity of nonlinear rotor systems, offering the following advantages:

[0091] A. High solution accuracy: The calculation method of this application realizes the automatic updating and solution of the Jacobian matrix by adding imaginary part perturbation to the degrees of freedom, thus avoiding human error;

[0092] B. High stability: The calculation method of this application does not depend on the selection of the perturbation step size, and can achieve accurate solution of response and sensitivity under any perturbation step size;

[0093] C. High solution efficiency: The calculation method of this application can simultaneously solve for the response, each order sensitivity, and the mixed sensitivity of each parameter as needed, thus achieving simultaneous solution and improving solution efficiency.

[0094] In addition to the purposes, features, and advantages described above, this application has other purposes, features, and advantages. A further detailed description of this application will be provided below with reference to the figures. Attached Figure Description

[0095] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application.

[0096] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort, wherein:

[0097] Figure 1This is a flowchart illustrating the method for calculating the dynamic response and sensitivity of a nonlinear rotor system according to a preferred embodiment of this application.

[0098] Figure 2 This is a schematic diagram of the finite element model of a single-disc rotor according to a preferred embodiment of this application;

[0099] Figure 3 This is a schematic diagram of the response calculation results of a preferred embodiment of this application;

[0100] Figure 4 This is a schematic diagram of the first-order sensitivity calculation results of a preferred embodiment of this application;

[0101] Figure 5 This is a schematic diagram of the second-order sensitivity calculation results of a preferred embodiment of this application;

[0102] Figure 6 This is a flowchart illustrating a method for calculating the dynamic response and sensitivity of a nonlinear rotor system according to another preferred embodiment of this application.

[0103] Figure 7 This is a schematic diagram of the module of the nonlinear rotor system dynamic response and sensitivity calculation device according to a preferred embodiment of this application;

[0104] Figure 8 This is a schematic diagram of a submodule of the module for solving the extended dimension of the equations of motion;

[0105] Figure 9 This is a schematic diagram of a submodule of the response and sensitivity solving module;

[0106] Figure 10 This is a schematic block diagram of an electronic device according to a preferred embodiment of this application;

[0107] Figure 11 This is a schematic diagram of the internal structure of a computer device according to a preferred embodiment of this application. Detailed Implementation

[0108] To better understand the technical solution of this application, a detailed description will be provided below in conjunction with the accompanying drawings and specific implementation methods.

[0109] It should be noted that the executing entity in this embodiment can be a computing service system with data processing, network communication, and program execution functions, such as a tablet computer, personal computer, or mobile phone, or a nonlinear rotor system dynamic response and sensitivity calculation device capable of performing the above functions. The following description uses a nonlinear rotor system dynamic response and sensitivity calculation device as the executing entity to illustrate this embodiment and the subsequent embodiments.

[0110] like Figure 1As shown, to address the aforementioned technical problems, a preferred embodiment of this application provides a method for calculating the dynamic response and sensitivity of a nonlinear rotor system, including the following steps:

[0111] S1. Based on the structural characteristics of the nonlinear rotor system, perform dynamic modeling of the nonlinear rotor system and establish the motion equations of the nonlinear rotor system;

[0112] S2. Establish the hypercomplex form of the motion equations of the nonlinear rotor system to obtain the hypercomplex motion equations of the nonlinear rotor system. Use complex matrix expression to expand the dimension of the motion equations of the nonlinear rotor system of complex matrix into an equivalent real matrix.

[0113] S3. Solve the hypercomplex equations of motion of the established nonlinear rotor system, extract the real and imaginary parts of the calculation results, and obtain the response of the nonlinear rotor system and the corresponding first-order and second-order sensitivities or mixed sensitivities.

[0114] This embodiment uses a single-disc rotor as an example. For details of the rotor system, please refer to [link to documentation]. Figure 2 As shown, the paper elaborates on the response of the nonlinear rotor system and the calculation methods for its first and second-order sensitivities to the nonlinear stiffness of the support.

[0115] Compared with existing technologies, the method for calculating the dynamic response and sensitivity of a nonlinear rotor system provided in this embodiment has the following advantages:

[0116] High solution accuracy: The calculation method in this embodiment achieves automatic updating and solution of the Jacobian matrix by adding imaginary perturbation to the degrees of freedom, thus avoiding human error.

[0117] High stability: The calculation method in this embodiment does not depend on the selection of the perturbation step size, and can achieve accurate solutions for response and sensitivity under any perturbation step size.

[0118] High solution efficiency: The calculation method in this embodiment can simultaneously solve for the response, each order sensitivity, and the mixed sensitivity of each parameter as needed, thus achieving simultaneous solution and improving solution efficiency.

[0119] Preferably, in step S1, when performing dynamic modeling of the nonlinear rotor system based on its structural characteristics, the system's mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear force vector F are obtained. nl Resultant external force vector F u Establish the equations of motion for the nonlinear rotor system:

[0120] ;

[0121] in, , and Let these represent the system's acceleration, velocity, and displacement vectors, respectively. This indicates the system rotational speed.

[0122] In this embodiment, in step S1, when performing dynamic modeling of the nonlinear rotor system based on its structural characteristics, the system's mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear force vector F are obtained. nl Resultant external force vector F u Finally, the equations of motion for the nonlinear rotor system are established. The advantages include: it can not only accurately describe the linear physical characteristics of the system such as inertia, elasticity, damping and gyroscopic effect, but also decouple the linear and nonlinear parts by separating the nonlinear terms into independent force vectors, thus laying the foundation for efficient solution by numerical integration. It also makes the mechanism analysis and working condition simulation of specific nonlinear factors clearer and more flexible.

[0123] Preferably, step S2 specifically includes the following steps:

[0124] S21. Add imaginary perturbations to the response and design parameters respectively, where the added perturbations are along different imaginary directions, and the unit of the imaginary direction is . , We obtain the hypercomplex form of the system's mass matrix M, stiffness matrix K, damping matrix C, and gyroscope matrix G;

[0125] S22. Calculate the nonlinear force using the response and parameters after adding the perturbation, and obtain the nonlinear restoring force F. nl hypercomplex form;

[0126] S23. Combine the mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear restoring force F. nl The resultant external force vector F u Substituting the hypercomplex form of the equations of motion into the equations of motion of the nonlinear rotor system, we obtain the hypercomplex form of the equations of motion of the nonlinear rotor system:

[0127] ;

[0128] in, , , , , and These are the hypercomplex forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, nonlinear force vector, and resultant external force vector, respectively, after adding imaginary perturbations along the imaginary direction;

[0129] S24. The parameters in the obtained hypercomplex form of the nonlinear rotor system's motion equations are equivalently represented in the real-field matrix form to obtain the dimension-expanded equivalent form of the motion equations:

[0130] ;

[0131] in , , , , and These are the equivalent matrix forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, eccentric force vector, and nonlinear force vector of the multi-complex domain after perturbation.

[0132] In this embodiment, step S2 specifically involves adding imaginary perturbations to the response and design parameters sequentially through steps S21 to S24 to obtain the nonlinear restoring force F. nl The method yields the hypercomplex form of the equations of motion for the nonlinear rotor system, and the equivalent form of the expanded-dimensional equations of motion. Its advantages include: cleverly transforming the original nonlinear dynamics problem in the real domain into an expanded-dimensional system in the hypercomplex domain, thus enabling simultaneous solutions to the system's dynamic response and its sensitivity to design parameters; avoiding the truncation error introduced by the difficulty in selecting the step size in the traditional finite difference method, significantly improving the accuracy and computational efficiency of sensitivity analysis, and providing accurate and efficient derivative support for subsequent rotor system optimization design, robustness assessment, and parameter identification.

[0133] Preferably, step S3 specifically includes the following steps:

[0134] S31. Determine the residual R of the equation, and use R... * To express the hypercomplex form of this residual, using Newmark-β The method is used to solve the equation of motion, and then x is... * n The value of x * n+1 The calculation yields the following formula, which determines the residual R of the equation of motion. * :

[0135] ;

[0136] In the formula, Let be the hypercomplex form of the equivalent resultant external force vector at the next moment. The equivalent stiffness is in hypercomplex form:

[0137] ;

[0138] ;

[0139] In the formula, for Newmark-β The integration constants are typically taken as 0.25 and 0.5.

[0140] S32. Determine the Jacobian matrix of the equation using J. * Let J be the hypercomplex form of this Jacobian matrix. The Jacobian matrix J can be determined according to the following formula. * :

[0141] ;

[0142] In the formula, Indicates the response x at the th k The residuals of the equations after adding imaginary perturbations to each degree of freedom express The k One degree of freedom along The coefficient in the imaginary part direction;

[0143] Using complex matrix representation, the Jacobian matrix is ​​expanded to an equivalent real matrix. The Jacobian matrix after dimension expansion with added imaginary perturbation is represented as follows:

[0144] ;

[0145] S33, Utilizing Equation Residuals And Jacobi matrix The equations are calculated to obtain the Jacobian matrix. after, The remaining value is represented as Using a complex matrix representation, this matrix is ​​expanded to an equivalent real matrix as shown below. This represents the real matrix after dimension expansion of the new residual vector following the addition of the imaginary part perturbation, as shown below:

[0146] ;

[0147] S34. Repeat the above steps to perform Newton's iterative calculation of the equations of motion according to the following formula until the hypercomplex form of the solution to the equations of motion is obtained:

[0148] ;

[0149] in, This is the response value from the previous moment. Let Jacobian matrix be the value from the previous time step. The remaining value of the Jacobian matrix calculated using the residual R at the previous time step;

[0150] S35. Extract the calculated response x j+1 The values ​​of the real and imaginary coefficients are used to obtain the response of the nonlinear rotor system (see...). Figure 3 ) and the corresponding first-order and second-order sensitivities or mixed sensitivities.

[0151] In this embodiment, step S3 specifically involves determining the residual R of the equation, determining the Jacobian matrix of the equation, obtaining the new residual, obtaining the hypercomplex form of the solution to the motion equation, and the response x in steps S31 to S35, in sequence. j+1 The values ​​of the real and imaginary coefficients yield the response and corresponding sensitivity, which has the following advantages: First, it can solve for both the response and sensitivity simultaneously; second, the second-order convergence property of Newton's method ensures efficiency; third, the construction of residuals and Jacobians makes the solution universal; and fourth, it can ultimately separate the real and imaginary information to meet engineering requirements.

[0152] Preferably, step S35 specifically includes the following steps:

[0153] S351. Extract the calculated response x j+1 Real part:

[0154] x=Re(x j+1 );

[0155] S352, Extracting response x j+1 Imaginary part coefficients combined with response x j+1 The real part is used to obtain the response and first- and second-order sensitivity results of the rotor system:

[0156] ;

[0157] ;

[0158] in, Indicates the dynamic response for design parameters First-order sensitivity (see) Figure 4 ), Indicates the dynamic response for design parameters Second-order sensitivity (see) Figure 5 ), Indicates the dynamic response x j+1 Along the virtual part The coefficient of direction, Indicates the dynamic response x j+1 Along the imaginary part of the mixture The coefficient of direction.

[0159] In this embodiment, step S35 is specifically implemented through steps S351 to S352 to extract the calculated response x. j+1The purpose of obtaining the values ​​of the real and imaginary coefficients is to obtain the response of the nonlinear rotor system and the corresponding sensitivity or hybrid sensitivity of each order. The advantages include: achieving an accurate mapping from the supercomplex domain calculation results to the physically understandable real domain response, where the real part directly corresponds to the actual vibration response of the system, and the imaginary coefficients contain the sensitivity information of each order of design parameters; this extraction process can be completed through algebraic operations without additional numerical differentiation or post-processing calculations, ensuring the integrity and homogeneity of the sensitivity information and avoiding information loss.

[0160] like Figure 6 As described above, another preferred embodiment of this application provides a method for calculating the dynamic response and sensitivity of a nonlinear rotor system, which can realize the calculation of the first-order sensitivity and response of a nonlinear rotor system.

[0161] like Figure 7 As shown, another preferred embodiment of this application also provides a device for calculating the dynamic response and sensitivity of a nonlinear rotor system, including:

[0162] The equation of motion module is used to perform dynamic modeling of the nonlinear rotor system based on its structural characteristics and to establish the equation of motion for the nonlinear rotor system.

[0163] The module for solving the extended dimension of the equations of motion is used to establish the hypercomplex form of the equations of motion of a nonlinear rotor system. It obtains the hypercomplex equations of motion of the nonlinear rotor system using complex matrix expression, and expands the dimension of the complex matrix equations of motion of the nonlinear rotor system into an equivalent real matrix.

[0164] The response and sensitivity solution module is used to solve the hypercomplex motion equations of the established nonlinear rotor system, extract the real and imaginary parts of the calculation results, and obtain the response of the nonlinear rotor system and the corresponding sensitivity of each order or mixed sensitivity.

[0165] The nonlinear rotor system dynamic response and sensitivity calculation device provided in this embodiment adopts the nonlinear rotor system dynamic response and sensitivity calculation method in the above embodiments, solving the technical problems of low calculation efficiency, insufficient calculation accuracy, and inaccurate results of existing dynamic response and sensitivity calculation methods. Compared with the prior art, the beneficial effects of the nonlinear rotor system dynamic response and sensitivity calculation device provided in this embodiment are the same as those of the nonlinear rotor system dynamic response and sensitivity calculation method provided in the above embodiments, and other technical features in the nonlinear rotor system dynamic response and sensitivity calculation device are the same as those disclosed in the methods of the above embodiments, and will not be repeated here.

[0166] like Figure 8 As shown, in another preferred embodiment of this application, the module for solving the extended dimension of the equations of motion specifically includes:

[0167] The first hypercomplex form determination module is used to add imaginary perturbations to the response and design parameters, respectively. The added perturbations are along different imaginary directions, with units of . , We obtain the hypercomplex form of the system's mass matrix M, stiffness matrix K, damping matrix C, and gyroscope matrix G;

[0168] The second hypercomplex form determination module is used to calculate the nonlinear force using the response and parameters after adding the perturbation, and obtain the nonlinear restoring force vector F. nl hypercomplex form;

[0169] The module for determining the hypercomplex form of the equations of motion is used to determine the mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear restoring force vector F. nl Resultant external force vector F u Substituting the hypercomplex form of the equations of motion into the equations of motion of the nonlinear rotor system, we obtain the hypercomplex form of the equations of motion of the nonlinear rotor system:

[0170] ;

[0171] in, , , , , and These are the hypercomplex forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, nonlinear force vector, and resultant external force vector, respectively, after adding imaginary perturbations along the imaginary direction;

[0172] The equivalence matrix determination module is used to equivalence each parameter in the obtained hypercomplex form of the nonlinear rotor system's motion equations using matrix form in the real field, thereby obtaining the dimension-expanded equivalent form of the motion equations:

[0173] ;

[0174] in , , , , and These are the equivalent matrix forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, eccentric force vector, and nonlinear force vector of the multi-complex domain after perturbation.

[0175] In this embodiment, the motion equation dimension expansion solution module specifically includes a first hypercomplex form determination module, a second hypercomplex form determination module, a motion equation hypercomplex form determination module, and an equivalent matrix determination module. These modules work together to obtain the equivalent form of the motion equation after dimension expansion. The advantages include: clear decoupling of complex mathematical transformations through modular functional division—the first and second modules are responsible for introducing hypercomplex perturbations to embed sensitivity information, the third module establishes the motion equation in the hypercomplex domain, and the fourth module organizes it into a real-domain dimension expansion matrix form that facilitates numerical solution; this hierarchical processing not only reduces the implementation difficulty and code maintenance cost of the algorithm but also makes the entire dimension expansion process highly scalable, flexibly adaptable to different types of nonlinear force models or multi-parameter sensitivity analysis requirements, and provides a structured mathematical foundation for subsequent efficient and accurate iterative solutions.

[0176] like Figure 9 As shown, in another preferred embodiment of this application, the response and sensitivity solving module specifically includes:

[0177] The residual determination module for the equations of motion is used to determine the residual R of the equations, using R... * To express the hypercomplex form of this residual, using Newmark-β The method is used to solve the equation of motion, and then x is... * n The value of x * n+1 The calculation yields the following formula, which determines the residual R of the equation of motion. * :

[0178] ;

[0179] In the formula, Let be the hypercomplex form of the equivalent resultant external force vector at the next moment. The equivalent stiffness is in hypercomplex form:

[0180] ;

[0181] ;

[0182] In the formula, for Newmark-β The integration constants are typically taken as 0.25 and 0.5.

[0183] The Jacobian matrix determination module is used to determine the Jacobian matrix of the equation, using J... * Let J be the hypercomplex form of this Jacobian matrix. The Jacobian matrix J can be determined according to the following formula. * :

[0184] ;

[0185] In the formula, Indicates the response x at the th k The residuals of the equations after adding imaginary perturbations to each degree of freedom express The k One degree of freedom along The coefficient in the imaginary part direction;

[0186] Using complex matrix representation, the Jacobian matrix is ​​expanded to an equivalent real matrix. The Jacobian matrix after dimension expansion with added imaginary perturbation is represented as follows:

[0187] ;

[0188] The residual update module is used to utilize the equation residuals. And Jacobi matrix The equations are calculated to obtain the Jacobian matrix. after, The remaining value is represented as Using a complex matrix representation, this matrix is ​​expanded to an equivalent real matrix as shown below. This represents the real matrix after dimension expansion of the new residual vector following the addition of the imaginary part perturbation, as shown below:

[0189] ;

[0190] The module for determining the hypercomplex form of the response is used to repeat the aforementioned steps, performing Newton's iterative calculations of the equations of motion according to the following formula, until the hypercomplex form of the solution to the equations of motion is obtained:

[0191] ;

[0192] in, This is the response value from the previous moment. Let Jacobian matrix be the value from the previous time step. The remaining value of the Jacobian matrix calculated using the residual R at the previous time step;

[0193] The response and sensitivity calculation module is used to extract the values ​​of the real and imaginary coefficients of the calculated response x, and obtain the response of the nonlinear rotor system and the corresponding sensitivity of each order or mixed sensitivity.

[0194] In this embodiment, the motion equation dimension expansion solution module specifically includes a residual determination module, a Jacobian matrix determination module, a residual update module, a hypercomplex form determination module for the response, and a response and sensitivity calculation module. These modules work together to obtain the response of the nonlinear rotor system and the corresponding sensitivities of various orders or mixed sensitivities. Its advantages include: constructing a complete, closed-loop iterative solution architecture—first, the residual determination module quantifies the solution deviation of the nonlinear equation; then, the Jacobian matrix determination module provides the acceleration direction for iterative convergence; and finally, the residual update module gradually corrects the solution, ultimately... The module for determining the hypercomplex form of the response and the module for calculating the response and sensitivity complete the accurate extraction of response and sensitivity information from the hypercomplex domain solution to the physical real domain. This modular process not only breaks down the complex nonlinear iterative solution process into functionally clear and easily implemented sub-tasks, improving the maintainability and portability of the algorithm, but also achieves the one-time acquisition of system response and sensitivity information while ensuring high-precision convergence through the embedded Jacobian matrix acceleration mechanism and synchronous solution strategy. This significantly improves the overall computational efficiency and provides complete and reliable data support for the subsequent dynamic design and robustness optimization of the rotor system.

[0195] like Figure 10 As shown, a preferred embodiment of this embodiment also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the nonlinear rotor system dynamic response and sensitivity calculation method in the above embodiment.

[0196] This embodiment also provides an electronic device that uses the nonlinear rotor system dynamic response and sensitivity calculation method in the above embodiments to solve the technical problems of low calculation efficiency, insufficient calculation accuracy and inaccurate results of existing dynamic response and sensitivity calculation methods. Compared with the prior art, the beneficial effects of the electronic device provided in this embodiment are the same as those of the nonlinear rotor system dynamic response and sensitivity calculation method provided in the above embodiments, and other technical features in the electronic device are the same as those disclosed in the methods of the above embodiments, which will not be repeated here.

[0197] like Figure 11 As shown, a preferred embodiment of this application also provides a computer device, which may be a terminal or a liveness detection server, and its internal structure diagram may be as follows. Figure 11As shown, the computer device includes a processor, memory, and a network interface connected via a system bus. The processor provides computational and control capabilities. The memory includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores the operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. The network interface is used to communicate with other external computer devices via a network connection. When the computer program is executed by the processor, it implements the steps of the aforementioned method for calculating the dynamic response and sensitivity of the nonlinear rotor system.

[0198] Those skilled in the art will understand that Figure 11 The structure shown is merely a block diagram of a portion of the structure related to the solution of this embodiment, and does not constitute a limitation on the computer device to which the solution of this embodiment is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have different component arrangements.

[0199] The computer device provided in this embodiment adopts the nonlinear rotor system dynamic response and sensitivity calculation method in the above embodiment, which solves the technical problems of low calculation efficiency, insufficient calculation accuracy and inaccurate results of the existing dynamic response and sensitivity calculation methods. Compared with the prior art, the beneficial effects of the computer device provided in this embodiment are the same as the beneficial effects of the nonlinear rotor system dynamic response and sensitivity calculation method provided in the above embodiment. In addition, other technical features in the electronic device are the same as the features disclosed in the method of the above embodiment, and will not be repeated here.

[0200] A preferred embodiment of this application also provides a storage medium, the storage medium including a stored program, which, when the program is executed, controls the device where the storage medium is located to perform the steps of the nonlinear rotor system dynamic response and sensitivity calculation method in the above embodiments.

[0201] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0202] If the functions described in this embodiment are implemented as software functional units and sold or used as independent products, they can be stored in one or more computing device-readable storage media. Based on this understanding, the parts of this embodiment that contribute to the prior art or the technical solution can be embodied in the form of a software product. This software product is stored in a storage medium and includes several instructions to cause a computing device (which may be a personal computer, server, mobile computing device, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this embodiment. The aforementioned storage media include: USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, optical disks, and other media capable of storing program code.

[0203] Those skilled in the art will understand that the embodiments of this example can be provided as methods, systems, or computer program products. Therefore, this example can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this example can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code. The solutions in this example can be implemented using various computer languages, such as the object-oriented programming language C++ and the embedded programming language C.

[0204] This embodiment is described with reference to flowchart illustrations and / or block diagrams of the method, apparatus (system), and computer program product according to this embodiment. It should be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations. Figure 1 One or more processes and / or boxes Figure 1 A system that specifies functions in one or more boxes.

[0205] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including an instruction set implemented in a process. Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0206] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0207] This embodiment also provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the above-described method for calculating the dynamic response and sensitivity of a nonlinear rotor system.

[0208] The computer program product provided in this embodiment solves the technical problems of low computational efficiency, insufficient computational accuracy, and inaccurate results in existing dynamic response sensitivity calculation methods. Compared with the prior art, the beneficial effects of the computer program product provided in this embodiment are the same as those of the nonlinear rotor system dynamic response and sensitivity calculation method provided in the above embodiments, and will not be repeated here.

[0209] This application has been successfully applied to the AES100 engine, providing crucial technical support for its smooth development. After applying this method and technology, the AES100 engine passed multiple full-engine bench tests, accelerated endurance tests, and full-engine verification under conditions such as high-altitude test benches and field flight tests, as well as in-flight flight tests. The verification results show that the rotor system designed using this application has a reasonable structure and stable dynamic characteristics.

[0210] Obviously, those skilled in the art can make various modifications and variations to this embodiment without departing from the spirit and scope of this embodiment. Therefore, if these modifications and variations of this embodiment fall within the scope of the claims of this embodiment and their equivalents, this embodiment is also intended to include these modifications and variations.

Claims

1. A method for calculating dynamic response and sensitivity of a nonlinear rotor system, characterized in that, Including the following steps: S1. Based on the structural characteristics of the nonlinear rotor system, perform dynamic modeling of the nonlinear rotor system and establish the motion equations of the nonlinear rotor system; S2. Establish the hypercomplex form of the motion equations of the nonlinear rotor system to obtain the hypercomplex motion equations of the nonlinear rotor system. Use complex matrix expression to expand the dimension of the complex matrix motion equations of the nonlinear rotor system into equivalent real matrix. The specific steps include: S21. Add imaginary perturbations to the response and design parameters respectively, where the added perturbations are along different imaginary directions, and the unit of the imaginary direction is . , We obtain the hypercomplex form of the system's mass matrix M, stiffness matrix K, damping matrix C, and gyroscope matrix G; S22, calculate the nonlinear force F using the response with added disturbance and the parameters nl , obtain the hypercomplex form of the nonlinear restoring force F nl ​ S23. Combine the mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear restoring force vector F. nl Resultant external force vector F u Substituting the hypercomplex form of the equations of motion into the equations of motion of the nonlinear rotor system, we obtain the hypercomplex form of the equations of motion of the nonlinear rotor system: ； in, , , , , and These are the hypercomplex forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, nonlinear force vector, and resultant external force vector, respectively, after adding imaginary perturbations along the imaginary direction; S24. The parameters in the obtained hypercomplex form of the nonlinear rotor system's motion equations are equivalently represented in the real-field matrix form to obtain the dimension-expanded equivalent form of the motion equations: ； in , , , , and These are the equivalent matrix forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, eccentric force vector, and nonlinear force vector of the multi-complex domain after perturbation; S3. Solve the established hypercomplex equations of motion for the nonlinear rotor system, extract the real and imaginary parts of the calculation results, and obtain the response of the nonlinear rotor system and the corresponding sensitivities of each order or mixed sensitivities. Specific steps include: S31. Determine the residual R of the equation, and use R... * To express the hypercomplex form of this residual, using Newmark-β The method is used to solve the equation of motion, and then x is... * n The value of x * n+1 The calculation yields the following formula, which determines the residual R of the equation of motion. * : ； In the formula, Let be the hypercomplex form of the equivalent resultant external force vector at the next moment. The equivalent stiffness is in hypercomplex form: ； ； wherein is Newmark-β the integration constant in the equation above, taken as 0.25 and 0.5; S32, determine the Jacobian matrix of the equation, using J * to represent the hypercomplex form of this Jacobian matrix, the Jacobian matrix J can be determined according to the following formula * : ； In the formula, Indicates the response x at the th k The residuals of the equations after adding imaginary perturbations to each degree of freedom express The k One degree of freedom along The coefficient in the imaginary part direction; Using complex matrix representation, the Jacobian matrix is ​​expanded to an equivalent real matrix. The Jacobian matrix after dimension expansion with added imaginary perturbation is represented as follows: ； S33, Utilizing Equation Residuals And Jacobi matrix The equations are calculated to obtain the Jacobian matrix. after, The remaining value is represented as Using a complex matrix representation, this matrix is ​​expanded to an equivalent real matrix as shown below. This represents the real matrix after dimension expansion of the new residual vector following the addition of the imaginary part perturbation, as shown below: ； S34. Repeat the above steps to perform Newton's iterative calculation of the equations of motion according to the following formula until the hypercomplex form of the solution to the equations of motion is obtained: ； in, This is the response value from the previous moment. Let Jacobian matrix be the value from the previous time step. The remaining value of the Jacobian matrix calculated using the residual R at the previous time step; S35. Extract the values ​​of the real and imaginary coefficients of the calculated response x to obtain the response of the nonlinear rotor system and the corresponding sensitivities of each order or mixed sensitivities. Specific steps include: S351, extract the calculated response x real part: x = Re(x j+1 ); S352, extract response x j+1 imaginary coefficients combine response x j+1 real part, get response of the rotor system and first order second order sensitivity results: ； ； in, Indicates the dynamic response for design parameters First-order sensitivity, Indicates the dynamic response for design parameters Second-order sensitivity, Indicates the dynamic response x j+1 Along the virtual part The coefficient of direction, The dynamic response x represents the mixed imaginary part. The coefficient of direction.

2. The method of claim 1, wherein, In step S1, based on the structural characteristics of the nonlinear rotor system, when performing dynamic modeling of the nonlinear rotor system, the system's mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear force vector F are obtained. nl Resultant external force vector F u Establish the equations of motion for the nonlinear rotor system: ； where, , and denote the acceleration, velocity and displacement vectors of the system, respectively, denotes the rotational speed of the system.

3. A device for calculating dynamic response and sensitivity of a nonlinear rotor system, characterized by, include: The equation of motion module is used to perform dynamic modeling of the nonlinear rotor system based on its structural characteristics and to establish the equation of motion for the nonlinear rotor system. The module for expanding the dimensions of the equations of motion is used to establish the hypercomplex form of the equations of motion for a nonlinear rotor system. It obtains the hypercomplex equations of motion for the nonlinear rotor system using complex matrix representation, expanding the dimensions of the complex matrix equations of motion for the nonlinear rotor system into equivalent real matrix equations. Specifically, it includes: The first hypercomplex form determination module is used to add imaginary perturbations to the response and design parameters, respectively. The added perturbations are along different imaginary directions, with units of . , We obtain the hypercomplex form of the system's mass matrix M, stiffness matrix K, damping matrix C, and gyroscope matrix G; The second hypercomplex form determination module is used to calculate the nonlinear force using the response and parameters after adding the perturbation, and obtain the nonlinear restoring force F. nl hypercomplex form; The module for determining the hypercomplex form of the equations of motion is used to determine the mass matrix M, stiffness matrix K, damping matrix C, gyroscope matrix G, and nonlinear restoring force F. nl Resultant external force vector F u Substituting the hypercomplex form into the equations of motion of the nonlinear rotor system, we obtain the hypercomplex form of the equations of motion of the nonlinear rotor system: ； in, , , , , and These are the hypercomplex forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, nonlinear force vector, and resultant external force vector, respectively, after adding imaginary perturbations along the imaginary direction; The equivalence matrix determination module is used to equivalence each parameter in the obtained hypercomplex form of the nonlinear rotor system's motion equations using matrix form in the real field, thereby obtaining the dimension-expanded equivalent form of the motion equations: ； in , , , , and These are the equivalent matrix forms of the mass matrix, damping matrix, gyroscope matrix, stiffness matrix, eccentric force vector, and nonlinear force vector of the multi-complex domain after perturbation; The response and sensitivity solution module is used to solve the hypercomplex equations of motion of the established nonlinear rotor system, extract the real and imaginary parts of the calculation results, and obtain the response of the nonlinear rotor system and the corresponding sensitivities of various orders or mixed sensitivities. Specifically, it includes: The residual determination module for the equations of motion is used to determine the residual R of the equations, using R... * To express the hypercomplex form of this residual, using Newmark-β The method is used to solve the equation of motion, and then x is... * n The value of x * n+1 The calculation yields the following formula, which determines the residual R of the equation of motion. * : ； wherein is the super-complex form of the equivalent resultant force vector at the next instant, is the super-complex form of the equivalent stiffness: ； ； wherein is Newmark-β the integration constant in the equation above, taken as 0.25 and 0.5; The Jacobian matrix determination module is used to determine the Jacobian matrix of the equation, using J... * Let J be the hypercomplex form of this Jacobian matrix. The Jacobian matrix J can be determined according to the following formula. * : ； In the formula, Indicates the response x at the th k The residuals of the equations after adding imaginary perturbations to each degree of freedom express The k One degree of freedom along The coefficient in the imaginary part direction; Using the matrix representation of complex numbers, the extended Jacobian matrix is converted to an equivalent real matrix, denoted as Jx, using represents the real matrix of the extended Jacobian matrix after adding the imaginary part perturbation, as follows: ； The residual update module is used to utilize the equation residuals. And Jacobi matrix The equations are calculated to obtain the Jacobian matrix. after, The remaining value is represented as Using a complex matrix representation, this matrix is ​​expanded to an equivalent real matrix as shown below. This represents the real matrix after dimension expansion of the new residual vector following the addition of the imaginary part perturbation, as shown below: ； The module for determining the hypercomplex form of the response is used to repeat the aforementioned steps, performing Newton's iterative calculations of the equations of motion according to the following formula, until the hypercomplex form of the solution to the equations of motion is obtained: ; in, This is the response value from the previous moment. Let Jacobian matrix be the value from the previous time step. The remaining value of the Jacobian matrix calculated using the residual R at the previous time step; The response and sensitivity calculation module is used to extract the values ​​of the real and imaginary coefficients of the calculated response x, obtaining the response of the nonlinear rotor system and the corresponding sensitivities of each order or mixed sensitivities. Specifically, it is used for: Extract the real part of the response x computed: x = Re(x j+1 ); extract the response x j+1 complex coefficients combine the response x j+1 real part, get the response of the rotor system and the first and second order sensitivity results: ； ； in, Indicates the dynamic response for design parameters First-order sensitivity, Indicates the dynamic response for design parameters The second-order sensitivity, Indicates the dynamic response x j+1 Along the imaginary part The coefficient of direction, The dynamic response x represents the mixed imaginary part. The coefficient of direction.

4. An electronic device, comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, when the processor executes the computer program, it implements a method for calculating the dynamic response and sensitivity of a nonlinear rotor system as described in any one of claims 1 to 2.

5. A computer-readable storage medium having stored thereon a computer program, characterized in that, When the computer program is executed by the processor, it implements the method for calculating the dynamic response and sensitivity of the nonlinear rotor system as described in any one of claims 1 to 2.

Images