High-fidelity modeling method, system and medium for gear-rotor system based on three-dimensional modal polycondensation and gyro equivalent mapping

Abstract

The application discloses a kind of high-fidelity modeling method, system and medium of gear-rotor system based on three-dimensional modal condensation and gyro equivalent mapping, belong to mechanical transmission and rotor dynamics field.The present application includes: establishing rotor shaft system three-dimensional entity model and cutting at step, establishing reference point and using continuous distribution coupling binding;Extract dimensionality three-dimensional stiffness matrix and mass matrix;Divide one-dimensional beam element, analytically calculate one-dimensional gyro matrix and after coordinate transformation zero padding mapping to system gyro array;Superimpose bearing cross stiffness damping and projected time-varying mesh stiffness;Solve dynamic response.The present application can overcome the defect of beam theory stiffness overestimation, bypass the extraction limit of commercial software gyro matrix, while ensuring modeling accuracy, improve the efficiency of gear-rotor system dynamics analysis.

Description

Technical Field

[0001] This invention relates to the fields of mechanical transmission, rotor dynamics, finite element simulation and digital twin technology, and in particular to a high-fidelity dimensionality reduction modeling and dynamic response prediction method, system and medium for gear-rotor-bearing systems that consider the three-dimensional local compliance effect of stepped shafts. Background Technology

[0002] Gear-rotor-bearing systems are widely used in high-end equipment such as aero engines, wind turbines, and electric drive axles for new energy vehicles. As modern machinery develops towards higher speeds, heavier loads, and lighter weights, the dynamic transmission error (DTE), vibration noise, and resonance avoidance design of gear systems have become a focus of engineering attention. The essence of gear transmission is strong torsional-bending coupling vibration, and the modeling accuracy of the rotor shaft system directly determines the accuracy of the overall gearbox dynamics prediction.

[0003] Currently, existing technologies face the following technical bottlenecks: Traditional one-dimensional (1D) beam theory suffers from the defect of "stiffness overestimation": Existing studies generally use lumped parameter methods or 1D Timoshenko beam elements to model the axis system. However, 1D beam theory is based on the "plane section assumption" and cannot capture the "three-dimensional local flexibility" and "non-planar warping effect" of stepped shafts at the abrupt change in diameter (step), resulting in an overestimation of torsional stiffness by 8%-10%, which leads to a serious overestimation of the predicted rotational speed of the torsional resonance peak and easily causes design misjudgments.

[0004] The computational cost of a three-dimensional (3D) solid model is extremely high: if a 3D solid finite element model is directly used for time-domain nonlinear gear dynamics calculation, the huge number of millions of degrees of freedom will lead to extremely time-consuming calculations, which cannot meet the real-time requirements of engineering frequency sweep analysis and digital twins.

[0005] Modal condensation in commercial software is limited by the extraction of the "gyroscopic effect": The Craig-Bampton (CB) modal condensation method is currently an effective means of reducing model dimensionality. However, in high-speed rotor dynamics, the gyroscopic effect caused by the Coriolis force cannot be ignored. The underlying mathematical architecture of mainstream commercial finite element software couples the gyroscopic effect into the asymmetric damping matrix, making it impossible to directly derive a pure gyroscopic matrix independent of the rotational speed, and easily leading to the divergence failure of substructure extraction algorithms based on real modes.

[0006] Therefore, there is an urgent need for a hybrid modeling method for gear-rotor dynamics that can preserve the complex local flexibility of a real 3D stepped axis, overcome the limitations of commercial software in extracting gyroscope matrices, and has extremely high computational efficiency. Summary of the Invention

[0007] This invention provides a high-fidelity modeling method, system, and medium for gear-rotor systems based on three-dimensional modal condensation and gyroscope equivalent mapping.

[0008] A high-fidelity modeling method for gear-rotor systems based on 3D modal condensation and gyroscope equivalent mapping includes the following steps: A three-dimensional solid finite element model of the gear rotor shaft system is established, and physical segmentation is performed at the stepped interface where the outer diameter of the shaft system changes abruptly. A reference point is established at the geometric center of each segmented section, and the reference point is bound to the corresponding section using continuous distributed coupling, thus giving the reference point a complete six degrees of freedom in space. Set up a substructure generation analysis step, set all reference points to retain physical master degrees of freedom, solve the internal modes of the fixed interface, and export the dimension-reduced three-dimensional stiffness matrix and three-dimensional mass matrix. High-fidelity stiffness and mass matrices were obtained using 3D solid Craig-Bampton modal condensation, while a one-dimensional global gyroscope matrix including polar rotational inertia was analytically calculated and assembled using one-dimensional Timoshenko beam theory. By mapping it to the global coordinate system of the 3D solid model through orthogonal coordinate transformation, the transformed one-dimensional gyroscope matrix is ​​obtained. ; Construct an all-zero system gyroscope matrix with the same dimensions as the three-dimensional mass matrix. The transformed one-dimensional gyroscope matrix is ​​then mapped using a zero-filled subspace. Write by degree of freedom index In the main physical degrees of freedom block, the gyroscope matrix block corresponding to the high-frequency internal mode is set to zero; Using the aforementioned three-dimensional stiffness matrix as the initial system stiffness matrix, an initial system damping matrix is ​​constructed; Based on the bearing mounting node and the gear meshing node, the bearing cross stiffness damping is superimposed on the initial system stiffness matrix and the initial system damping matrix to obtain the system stiffness matrix and the system damping matrix, and the time-varying gear meshing stiffness is projected onto the corresponding degree of freedom along the spatial meshing line. Based on the system stiffness matrix, the three-dimensional mass matrix, the system gyroscope matrix, and the system damping matrix, complex eigenvalue equations or dynamic differential equations incorporating gyroscope effects are constructed to solve the dynamic response of the gear-rotor system.

[0009] The above method uses 3D solid CB modal condensation to obtain high-fidelity stiffness and mass matrices, and combines them with the 1D Timoshenko beam analytical gyro matrix. The two are fused through orthogonal coordinate transformation and zero-fill subspace mapping, thereby effectively releasing the 3D local flexibility at the step of the stepped axis and overcoming the defect of overestimation of torsional stiffness in traditional beam theory. By equivalent embedding of the 1D gyro matrix and the 3D condensation matrix, the technical limitation that commercial software cannot directly derive pure gyro matrices is bypassed. Finally, the million-degree-of-freedom model is compressed to within 100 dimensions, significantly improving computational efficiency while ensuring high fidelity.

[0010] Furthermore, the continuous distributed coupling allows the cross-section to undergo Poisson contraction and non-planar warping under stress, thereby releasing the three-dimensional local compliance at the stepped shaft steps. This feature realistically simulates the local distortion of stress flow at the steps, corrects the errors of traditional models in torsional and tensile / compressive stiffness, and improves the modal frequency prediction accuracy by nearly 10%.

[0011] Furthermore, the one-dimensional global gyroscope matrix Based on Timoshenko beam theory, the scaling factor of the gyroscope matrix of a single-segment axis element... satisfy: ; In the formula, For density, It is the polar moment of inertia. For the unit length, The shear deformation parameter considering the Cowper coefficient is expressed as follows: ; in For elastic modulus, Shear modulus For the cross-sectional area, Let the moment of inertia of the cross section be... denoted as the cross-sectional shape shear coefficient. This feature utilizes an analytical method to accurately construct a pure gyroscope matrix independent of rotational speed, avoiding the problem of asymmetric matrix contraction and divergence found in commercial software.

[0012] Furthermore, the orthogonal coordinate transformation employs a transformation matrix. Achieve, satisfy: . This orthogonal coordinate transformation realizes the right-hand rule mapping between the local coordinate system of one-dimensional beam theory and the three-dimensional global coordinate system, ensuring the correct conversion of physical fields.

[0013] Furthermore, the step of setting the gyroscope matrix corresponding to the high-frequency internal modes to zero in blocks is achieved through zero-filling mapping, resulting in the final system gyroscope matrix. Represented as: ; in, Let be the dimension of the three-dimensional mass matrix. Based on the physical mechanism that the macroscopic angular momentum of the internal modes tends to zero, lossless embedding of the gyroscope effect is achieved while maintaining the positive definiteness of the matrix.

[0014] The present invention also provides a high-fidelity modeling system for gear-rotor systems based on three-dimensional modal condensation and gyroscope equivalent mapping for implementing the method, comprising: The solid model building module is used to create a three-dimensional solid finite element model of the gear rotor shaft system and cut it at the step interface; The reference point coupling module is used to establish a reference point at the center of the cutting section and to bind it using continuous distributed coupling. The condensation matrix extraction module is used to extract the dimension-reduced three-dimensional stiffness matrix and three-dimensional mass matrix based on the Craig-Bampton method, with the reference point as the retained master degree of freedom. The gyroscope matrix analysis module is used to calculate the one-dimensional global gyroscope matrix and perform coordinate transformations based on Timoshenko beam theory. The matrix fusion module is used to construct the all-zero system gyroscope matrix and map the transformed one-dimensional gyroscope matrix to the main physical degree of freedom blocks; The boundary condition application module is used to superimpose the bearing cross stiffness damping and the projected time-varying meshing stiffness of the gear. The dynamic response solver module is used to construct and solve complex eigenvalue equations or dynamic differential equations that include gyroscopic effects.

[0015] This system can achieve the above methods and has the same technical effect.

[0016] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of any of the methods described above.

[0017] Compared with the prior art, the present invention has the following beneficial effects: (a) Extremely high dynamic prediction accuracy; This invention utilizes a three-dimensional solid Craig-Bampton modal condensation and continuous distributed coupling technique. By establishing reference points at the abrupt change in the stepped interface of a stepped shaft and employing continuous distributed coupling, it effectively releases the local flexibility and Poisson effect of the solid model. This realistically captures the three-dimensional local flexibility and non-planar warping effects that traditional one-dimensional Timoshenko beam theory, due to its "plane section assumption," cannot reflect. Numerical verification shows that this method overcomes the stiffness hardening problem inherent in traditional one-dimensional models when calculating the torsional and tension / compression modes of stepped shafts, corrects nearly 10% of the modal frequency error, and avoids design misjudgments caused by the predicted offset of the torsional resonance peak position in actual gearbox operation.

[0018] (ii) Breaking through the limitations of the underlying commercial software to achieve precise embedding of the gyroscope effect; This invention proposes a subspace equivalent mapping technique between a one-dimensional gyroscope matrix and a three-dimensional condensed matrix. Addressing the limitation of mainstream commercial finite element software (such as Abaqus and Ansys) in directly deriving a pure gyroscope matrix independent of rotational speed from an asymmetric damping matrix, this invention utilizes the Timoshenko beam analytical method to construct a precise one-dimensional gyroscope matrix. Through orthogonal coordinate transformation and zero-filling subspace mapping, this matrix is ​​equivalently embedded into a three-dimensional reduced-dimensional matrix. This scheme reproduces the bifurcation phenomenon of the Coriolis force precession in high-speed rotors with minimal computational cost, bypasses the divergence problem during asymmetric matrix condensation, and achieves hybrid modeling with decoupled physical fields.

[0019] (iii) Significant improvement in computational efficiency; This invention compresses a three-dimensional solid contact nonlinear transient model with millions of degrees of freedom into a system of ordinary differential equations of less than 100 dimensions using the Craig-Bampton method (typically...). While ensuring that the three-dimensional local stress distortion characteristics are not lost, the Newmark-β method is used for time-domain integration, which can significantly improve the efficiency of transient response solution on ordinary workstations. Compared with the original three-dimensional solid model (which usually takes several hours), the computational efficiency is improved by several orders of magnitude, which can meet the real-time requirements of engineering frequency sweep analysis and digital twins.

[0020] (iv) A complete framework for solving nonlinear dynamics; This invention establishes a nonlinear dynamics solution framework for gear-rotor-bearing systems. It precisely assembles the dimensionality-reduced rotor super-element matrix with the nonlinear bearing cross stiffness and the time-varying gear meshing stiffness matrix projected onto the spatial meshing line. It can simultaneously perform critical speed prediction (Campbell plot) and transient response solution (including nonlinear forces such as tooth flank clearance), providing a high-fidelity and high-efficiency simulation tool for vibration and noise analysis and resonance avoidance design of high-end equipment. Attached Figure Description

[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort, wherein: Figure 1 A schematic diagram of the physical model coupling the three-dimensional solid reference points and the continuous distribution of the rotor stepped shaft; Figure 2 A comparison chart of the first 20 frequencies of cross-validation based on 3D solid CB condensation and one-dimensional Timoshenko beam theory; Figure 3 A bar chart showing the relative error of stiffness overestimation between the one-dimensional model and the three-dimensional high-fidelity model; Figure 4 A high-fidelity modeling diagram of a gear-rotor system based on 3D modal condensation and gyroscope equivalent mapping; Figure 5 Campbell diagram for a gear-rotor system considering gyroscopic mapping effects. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. These embodiments are only for explaining the invention and are not intended to limit it.

[0023] Example 1: 3D solid modeling and cross-sectional coupling of rotor shaft system like Figure 1 As shown, based on the actual dimensions of the gear rotor system (e.g., the input shaft of an electric drive axle in a new energy vehicle, with a total length of 300mm, a maximum diameter of 60mm, and three abrupt diameter changes), a solid model of the rotor is created using 3D modeling software (such as SolidWorks). Considering the local compliance effect of stress flow at the steps in the real physical world, sections are divided at all steps where the outer diameter changes along the axial direction of the model. In finite element preprocessing software (such as Abaqus / CAE), a circle is generated at the center of each section. Reference points Continuous distributed coupling constraints are used to define the reference point. It is bound to the surface of the solid cross-section. This coupling method allows the cross-section to undergo Poisson contraction and non-planar warping under stress, effectively releasing local flexibility; it truly reflects the three-dimensional stress distortion of the stepped shaft under torsion and tension / compression conditions, overcoming the overestimation of stiffness caused by the plane section assumption of the traditional one-dimensional beam model (see...). Figure 3 (Significantly corrects modal frequency errors).

[0024] Example 2: 3D Modal Condensation Matrix Extraction Based on the Craig-Bampton Method Define all The six spatial degrees of freedom (three translational and three rotational) at each reference point are the retained physical master degrees of freedom. A substructure generation analysis step is set up, and the Lanczos solver is invoked to calculate... The primary mode inside the fixed interface (usually taken as) (Can cover the target frequency band). The dimensionality-reduced three-dimensional stiffness matrix is ​​extracted. With the three-dimensional mass matrix The total dimensionality of the model after dimensionality reduction. Typically, it does not exceed 100 dimensions, and compared to the original million-level three-dimensional solid model, the computational efficiency is improved by several orders of magnitude. Figure 2 This study presents a comparison of the first 20 frequencies of cross-validation based on 3D solid CB condensation and one-dimensional Timoshenko beam theory. Figure 3 This quantifies the relative error of stiffness overestimation of the one-dimensional model compared to the three-dimensional high-fidelity model. Example 2 achieves efficient dimensionality reduction of the model while preserving three-dimensional local flexibility information.

[0025] Example 3: Analytical Derivation of One-Dimensional Gyroscope Matrix and Orthogonal Coordinate Transformation The gyroscopic effect depends only on the polar moment of inertia distribution of the cross section and is unaffected by local stress concentration. Based on Timoshenko beam theory, the gear rotor shaft system is divided into several one-dimensional beam elements according to the reference point position. For each element, its gyroscopic matrix scaling factor satisfies: ; in, Density (e.g., 7800 kg / m³) It is the polar moment of inertia. For the unit length, To account for the shear deformation parameters of the Cowper coefficient: ; In the formula, This refers to the elastic modulus (e.g., 2.06e11 Pa). Shear modulus For the cross-sectional area, Let the moment of inertia of the cross section be... The shear coefficient for the cross-sectional shape is 0.9 (0.9 for a solid circular cross-section). Assemble all elements to obtain the global one-dimensional gyroscope matrix. Since one-dimensional theory often uses the Z-axis as the axis, while three-dimensional finite element methods often use the X-axis, an orthogonal coordinate transformation matrix is ​​constructed. Implement right-hand rule mapping and perform coordinate transformation: ; This step obtains a pure gyroscope matrix independent of rotational speed through analytical methods, bypassing the technical limitation that commercial finite element software (such as Abaqus and Ansys) cannot directly extract the gyroscope matrix from the asymmetric damping matrix, and the transformed matrix is ​​consistent with the coordinate system of the three-dimensional model.

[0026] Example 4: Subspace Zero-Fill Mapping Fusion like Figure 4 As shown, this step is based on the following physical mechanism: Craig-Bampton polycondensation extraction The internal modes characterize local high-frequency micro-deformations under fixed cross-section conditions, with their overall macroscopic angular momentum approaching zero. Therefore, the coupling effect of the Coriolis force (gyroscopic effect) on the internal modes is negligible. The construction dimension is... All-zero system gyroscope matrix Using the zero-filling mapping method, G1D based on physical master degrees of freedom is transformed. G 1 D Matrix written directly In the upper left corner (physical degree of freedom block), the block corresponding to the high-frequency internal modes remains at zero: ; This zero-filling mapping maintains the consistency of the system matrix dimensions and the antisymmetric structural characteristics of the gyroscope matrix, while avoiding the destruction of the positive definiteness of the three-dimensional condensed mass matrix and stiffness matrix. Figure 4 This paper fully demonstrates the high-fidelity modeling process of a gear-rotor system based on 3D modal condensation and gyro equivalent mapping. This step achieves the equivalent embedding of the gyro matrix, which can reflect the bifurcation phenomenon of the forward and reverse precession of the high-speed rotor (see [link]). Figure 5 Campbell diagram).

[0027] Example 5: Applying boundary conditions to a gear-bearing nonlinear system Extract the system stiffness matrix quality matrix Import the computational software (such as MATLAB). Construct the initial system damping matrix. (This embodiment uses Rayleigh damping) ,in , Determined based on empirical modal damping ratio, for example , Based on the system installation topology, the cross stiffness matrix of the sliding bearing is superimposed on the radial degrees of freedom corresponding to the support nodes. and cross-damping matrix The final system damping moment is obtained. and system stiffness matrix .

[0028] For gear meshing, a spatial meshing line projection vector is constructed based on the pressure angle (e.g., 20°) and helix angle (e.g., 15°) of the gear pair. The one-dimensional time-varying meshing stiffness sequence (This can be obtained through finite element contact calculations or analytical formulas) The projection is the spatial coupling stiffness matrix. : ; Then Superimposed on the system's global stiffness matrix The corresponding main diagonal and the coupling cross term of the two gear nodes. Spatial coupling stiffness matrix. The time-varying and nonlinear characteristics of gear meshing are precisely introduced, providing a realistic excitation for subsequent transient response prediction.

[0029] Example 6: High-fidelity Campbell plots and transient dynamics solutions Critical speed prediction: Constructing a system that includes speed prediction Complex eigenvalue equations: ; Solve for the imaginary eigenvalues ​​at different speeds and plot the Campbell diagram (e.g.) Figure 5 As shown in the figure, it accurately captures the true torsional resonance speed of the forward and reverse precession bifurcation and gear system. Compared with the traditional one-dimensional model, the accuracy of resonance peak position prediction is greatly improved, avoiding design misjudgments caused by overestimation of model stiffness.

[0030] Nonlinear transient solution: Based on Newton's laws of motion, construct the ordinary differential dynamic equations of the gear rotor system: ; in External torque (such as the output torque of a motor). For nonlinear forces such as tooth flank clearance (e.g., using a piecewise linear clearance function). Newmark- β Law( , By performing step integration on the reduced-dimensional equation, the dynamic transmission error and vibration response of the gear system are obtained. In typical simulations, this reduced-dimensional model can significantly reduce the computational degrees of freedom, and compared with the original three-dimensional solid model (which takes several hours), it greatly improves the efficiency of solving transient responses while ensuring accuracy.

[0031] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A high-fidelity modeling method for gear-rotor systems based on three-dimensional modal condensation and gyro equivalent mapping, characterized in that, Includes the following steps: A three-dimensional solid finite element model of the gear rotor shaft system is established, and physical segmentation is performed at the stepped interface where the outer diameter of the shaft system changes abruptly. A reference point is established at the geometric center of each segmented section, and the reference point is bound to the corresponding section using continuous distributed coupling, thus giving the reference point a complete six degrees of freedom in space. Set up a substructure generation analysis step, set all reference points to retain physical master degrees of freedom, solve the internal modes of the fixed interface, and export the dimension-reduced three-dimensional stiffness matrix and three-dimensional mass matrix. High-fidelity stiffness and mass matrices were obtained using 3D solid Craig-Bampton modal condensation, while a one-dimensional global gyroscope matrix including polar rotational inertia was analytically calculated and assembled using one-dimensional Timoshenko beam theory. By mapping it to the global coordinate system of the 3D solid model through orthogonal coordinate transformation, the transformed one-dimensional gyroscope matrix is ​​obtained. ; Construct an all-zero system gyroscope matrix with the same dimensions as the three-dimensional mass matrix. The transformed one-dimensional gyroscope matrix is ​​then mapped using a zero-filled subspace. Write by degree of freedom index In the main physical degrees of freedom block, the gyroscope matrix block corresponding to the high-frequency internal mode is set to zero; Using the aforementioned three-dimensional stiffness matrix as the initial system stiffness matrix, an initial system damping matrix is ​​constructed; Based on the bearing mounting node and the gear meshing node, the bearing cross stiffness damping is superimposed on the initial system stiffness matrix and the initial system damping matrix to obtain the system stiffness matrix and the system damping matrix, and the time-varying gear meshing stiffness is projected onto the corresponding degree of freedom along the spatial meshing line. Based on the system stiffness matrix, the three-dimensional mass matrix, the system gyroscope matrix, and the system damping matrix, complex eigenvalue equations or dynamic differential equations incorporating gyroscope effects are constructed to solve the dynamic response of the gear-rotor system.

2. The method according to claim 1, characterized in that, The continuous distributed coupling allows the cross section to undergo Poisson contraction and non-planar warping under stress, effectively releasing the three-dimensional local compliance at the stepped shaft steps.

3. The method according to claim 1, characterized in that, The one-dimensional global gyroscope matrix Based on Timoshenko beam theory, the scaling factor of the gyroscope matrix of a single-segment axis element... satisfy: ; In the formula, For density, It is the polar moment of inertia. For the unit length, The shear deformation parameter considering the Cowper coefficient is expressed as follows: ; in For elastic modulus, Shear modulus For cross-sectional area, Let the moment of inertia of the cross section be... is the shear coefficient for cross-sectional shape.

4. The method according to claim 1, characterized in that, The orthogonal coordinate transformation employs a transformation matrix. Achieve, satisfy: 。 5. The method according to claim 1, characterized in that, The step of setting the gyroscope matrix corresponding to the high-frequency internal modes to zero in blocks is achieved through zero-filling mapping, resulting in the final system gyroscope matrix. Represented as: ; in, The dimension of the three-dimensional mass matrix is ​​given by the zero-filling mapping, which maintains the consistency of the system matrix dimensions and the antisymmetric structural characteristics of the gyroscope matrix, while avoiding the destruction of the positive definiteness of the three-dimensional condensed mass matrix and stiffness matrix.

6. A high-fidelity modeling system for gear-rotor systems based on three-dimensional modal condensation and gyro equivalent mapping for implementing the method described in any one of claims 1-5, characterized in that, include: The solid model building module is used to create a three-dimensional solid finite element model of the gear rotor shaft system and cut it at the step interface; The reference point coupling module is used to establish a reference point at the center of the cutting section and to bind it using continuous distributed coupling. The condensation matrix extraction module is used to extract the dimension-reduced three-dimensional stiffness matrix and three-dimensional mass matrix based on the Craig-Bampton method, with the reference point as the retained master degree of freedom. The gyroscope matrix analysis module is used to calculate the one-dimensional global gyroscope matrix and perform coordinate transformations based on Timoshenko beam theory. The matrix fusion module is used to construct the all-zero system gyroscope matrix and map the transformed one-dimensional gyroscope matrix to the main physical degree of freedom blocks; The boundary condition application module is used to superimpose the bearing cross stiffness damping and the projected time-varying meshing stiffness of the gear. The dynamic response solver module is used to construct and solve complex eigenvalue equations or dynamic differential equations that include gyroscopic effects.

7. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the method as described in any one of claims 1 to 5.

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