Three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis method and system

Abstract

The invention discloses a three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis method and system, and relates to the technical field of energy power, and the method comprises the steps: building an excitation rotor three-dimensional finite element dynamic model with an axial groove; uncertainty quantification is carried out on the excitation rotor system, and an excitation rotor system dynamical model containing interval uncertainty is established; solving the uncertainty steady-state response by adopting a Latin hypercube sampling method and based on a Kriging agent model; according to the method, interval uncertainty is considered, a dynamic response decoupling coordinate method is adopted to suppress false formants, an agent model based on Kriging and Latin hypercube sampling methods is combined, a rapid analysis method for the uncertainty steady-state response of the three-dimensional asymmetric rotor system is established, and the steady-state response of the three-dimensional asymmetric rotor system is analyzed; according to the method, the phenomenon that false formants are likely to occur when an interval method is used for asymmetric rotor steady-state response uncertainty analysis is solved.

Description

Technical Field

[0001] The present invention relates to the field of energy and power technology, and in particular to a method and system for analyzing uncertainty steady-state responses of a three-dimensional asymmetric excitation rotor. Background Art

[0002] To meet power generation requirements, the excitation rotor in a steam turbine unit is slotted axially, making its cross-section non-circular. This results in different moments of inertia about the two principal axes of inertia, leading to differences in bending stiffness—a phenomenon known as axial asymmetry. Typically, transverse arc slots or axial slots are incorporated into the large teeth to minimize this asymmetry and reduce vibration. In early designs, trial-and-error methods were often used to determine the slotting pattern, but a systematic and effective approach has yet to be established. This is particularly important in the design and manufacture of high-parameter, high-capacity steam turbine excitation rotors.

[0003] Previous studies on the dynamic characteristics of rotor systems have primarily employed deterministic methods, meaning that the rotor's geometric model parameters and boundary condition parameters, such as bearing supports, all have fixed values. Even when studying the impact of these parameters on the dynamic characteristics, these studies typically focus on a limited number of parameters and then investigate how the dynamic characteristics change as a function of these parameters. These studies can be considered deterministic.

[0004] However, uncertainty is widespread in reality. When multiple parameters are simultaneously uncertain, it will have a significant impact on the rotor dynamics. Therefore, it is necessary to conduct research on the quantification of rotor system uncertainty. The Monte Carlo method and the scanning method are two robust computational methods that are easy to operate. However, their disadvantage is that they require a large number of samples to ensure computational convergence, and the number of samples increases exponentially with the dimension of the uncertainty parameter, leading to the "curse of dimensionality" in high-dimensional cases, making calculations difficult. In addition, previous research on the quantification of rotor system uncertainty has only focused on simple models such as beam elements, and there have been few successful cases of application to three-dimensional rotor systems, especially asymmetric rotors. To overcome this problem, the polynomial chaos expansion method (PCE) and the polynomial interval method have been developed to establish surrogate models of random processes. However, these two methods require few samples and have fast computational speeds, but are prone to generating false resonance peaks, resulting in errors in the analysis results. Summary of the Invention

[0005] In view of the shortcomings of the existing technology in solving high-dimensional uncertainty problems such as the uncertainty quantification of the nonlinear steady-state response of large asymmetric rotors, which is difficult to converge and takes a long time, and is prone to false resonance peaks, resulting in errors in the analysis results, the present invention proposes a three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis method and system. By establishing a three-dimensional finite element dynamic model of the excitation rotor with axial slots, considering interval uncertainty, using the dynamic response decoupling coordinate method to suppress false resonance peaks, and combining with the proxy model based on Kriging and Latin hypercube sampling methods, a fast analysis method for the uncertainty steady-state response of the three-dimensional asymmetric rotor system is established, thereby solving the problems existing in the existing technology.

[0006] A method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor includes the following steps: Based on the three-dimensional finite element dynamic model of the three-dimensional asymmetric excitation rotor, the dynamic model of the excitation rotor system with interval uncertainty is established by introducing interval uncertainty parameters. The Latin hypercube sampling method is used to map the uncertain parameters in the dynamic model of the excitation rotor system to the standard hypercube space to generate a Latin hypercube sampling sample space. Response analysis is performed on each sample point in the sample space, and a deterministic steady-state response curve is established for each sample point. Extreme value detection is performed on the deterministic steady-state response curve, and the deterministic steady-state response curve is divided into multiple sub-segments based on the detected extreme value points. The speed and response amplitude are decoupled in the dynamic response coordinate to determine the decoupling coordinates of each sub-segment. Based on the decoupling coordinates of each sub-segment, the deterministic steady-state response curve is converted into an independent mapping relationship between the dynamic response coordinates and the speed and response amplitude. A Kriging model is established based on the uncertainty parameters and the independent mapping relationship after decoupling. Monte Carlo simulation is used to generate a new Latin hypercube sampling space. The mapping relationship between the dynamic response coordinates of each sample in the sample space and the speed and response amplitude is predicted through the established Kriging model, generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system. Based on the steady-state response envelope curve that characterizes the uncertainty of the excitation rotor system, the uncertainty steady-state response of the three-dimensional asymmetric excitation rotor is analyzed and obtained.

[0007] Furthermore, the three-dimensional finite element dynamic model based on the three-dimensional asymmetric excitation rotor introduces interval uncertainty parameters to establish an excitation rotor system dynamic model containing interval uncertainty, which specifically includes the following steps: Establish a three-dimensional finite element dynamic model of a three-dimensional asymmetric excitation rotor: ; Definition of the existence of excitation rotor system n uinterval uncertainty parameters, the interval vector is expressed as: ; Represent an interval vector as a median and interval radius: ; in: ; Where: is an interval vector; 、 are the lower and upper bound vectors of the interval; is the median of the interval vector; is the radius of the interval vector; According to the three-dimensional finite element dynamic model of the excitation rotor, the dynamic equation of the excitation rotor system with interval uncertainty is obtained as follows: ; in, is the mass matrix of the excitation rotor; is the Coriolis damping matrix of the excitation rotor; is the static stiffness matrix of the excitation rotor; Softening matrix for excitation rotor rotation; is the displacement response of the two-pole generator bearing system; is the velocity vector of the two-pole generator bearing system; is the acceleration vector of the two-pole generator bearing system; is the gravity vector; is the cosine modulation term of the bearing stiffness; is the sinusoidal modulation term of the bearing stiffness; , is the Coriolis damping matrix of the rotor, is the static stiffness matrix of the rotor, is the dynamic stiffness matrix of the rotor, is the static stiffness component of the bearing, is the fundamental angular frequency of the excitation.

[0008] Furthermore, the step of performing extreme value detection on the deterministic steady-state response curve and dividing the deterministic steady-state response curve into a plurality of sub-segments according to the detected extreme value points specifically includes the following steps: Mark the extreme points of the deterministic steady-state response curve and initialize the motion response decoupling coordinates of the starting point to 0; When traversing the trajectory of the deterministic steady-state response curve, the motion response decoupling coordinate value increases by 1 unit each time an extreme point is detected, and increases by 1 unit again when reaching the end point of the deterministic steady-state response curve; The deterministic steady-state response curve is divided into several sub-segments based on the identified starting points, extreme points, and end points.

[0009] Furthermore, the method uses the Latin hypercube sampling method to map the uncertain parameters in the dynamic model of the excitation rotor system to the standard hypercube space to generate the Latin hypercube sampling sample space; specifically, the following steps are included: The uncertainty parameter in the dynamic equation of the excitation rotor system with interval uncertainty is set to , each parameter follows a uniform distribution , mapping the parameters to the standard hypercube space , the normalized mapping is: ; in, a i 、 b i Refers to each uncertainty parameter Upper and lower limits; Divide the dimensional subspace into In each subinterval, a sample point is randomly selected and the order of intervals in each dimension is randomly arranged. The sample points of each dimension are randomly arranged and combined, and the number of samples is: ; in, is the truncation order of the surrogate model, is the uncertainty dimension; The sample points from Mapping back to the original parameter interval, we get: ; Finally, the Latin hypercube sampling sample point set is .

[0010] Furthermore, the step of performing response analysis on each sample point in the sample space and establishing a deterministic steady-state response curve for each sample point specifically includes the following steps: For each sample point , the complex exponential harmonic balance method is used to solve the steady-state response of the dynamic equation of the excitation rotor system with interval uncertainty: ; Assuming the steady-state response to be a periodic function, it can be expressed using a complex exponential series: ; in, is the order of the harmonic, is the harmonic cutoff order, is the complex amplitude of each order harmonic component in the response, is the fundamental angular frequency of the excitation, is the complex amplitude of each order harmonic component in the excitation; is the displacement response vector of the generator bearing system with respect to time; About time t The gravity vector of Substituting the expanded form of the steady-state response into the dynamic equation and matching the coefficients of each harmonic term, we obtain the linear equation system: ; in, is the mass matrix, that is, the harmonic order The inertia term, is the motion response solution; Solve for domain coefficients by matrix inversion or iterative methods , reconstruct the time domain response, and obtain the deterministic steady-state response curve of each sample point: ; in is the discrete number of speed points, is the speed value of each sample point of the response curve, The response amplitude of each sample point is .

[0011] Furthermore, the Kriging model is established based on the uncertainty parameters and the independent mapping relationship after decoupling, which specifically includes the following steps: Define input parameters as uncertainty parameters ,in express n dimensional real space, the output parameter is the length of each sub-segment trajectory Corresponding speed and amplitude ; Set the output parameter to follow the Gaussian process : ; in, is the covariance function that controls the speed, is the covariance function that controls the response amplitude, the mean function Take a constant or linear term, covariance function Using the Gaussian kernel, we get: ; Optimizing hyperparameters via maximum likelihood estimation ,in is the output variance, For then dimensional length scale, optimized hyperparameters for: ; in is the covariance matrix, which is represented by the covariance function calculate; is the training data, 、 、 is the mean function value; For each trajectory length , respectively train the Kriging model: ; .

[0012] Furthermore, the method uses Monte Carlo simulation to generate a new Latin hypercube sampling sample space, and uses the established Kriging model to predict the mapping relationship between the dynamic response coordinates of each sample in the sample space and the speed and response amplitude, thereby generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system. Specifically, the method includes the following steps: Generate new Latin hypercube sample sets using Monte Carlo simulation; The established Kriging model is used to predict the mapping relationship between the deterministic steady-state response curve trajectory length of each sample and the speed and response amplitude, and the speed and amplitude boundaries of each sample are determined. The speed and amplitude boundaries are mapped back to the original deterministic steady-state response curve according to the sub-segment trajectory length, and a steady-state response envelope curve is generated that can characterize the uncertainty of the excitation rotor system.

[0013] The present invention also includes a three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis system, comprising: The dynamic model construction module is used to establish the dynamic model of the excitation rotor system with interval uncertainty by introducing interval uncertainty parameters based on the three-dimensional finite element dynamic model of the three-dimensional asymmetric excitation rotor; A solution module is used to map the uncertain parameters in the dynamic model of the excitation rotor system to the standard hypercube space using the Latin hypercube sampling method, generate a Latin hypercube sampling sample space, perform response analysis on each sample point in the sample space, and establish a deterministic steady-state response curve for each sample point; perform extreme value detection on the deterministic steady-state response curve, and divide the deterministic steady-state response curve into multiple sub-segments based on the detected extreme value points; decouple the speed and response amplitude under the dynamic response coordinate to determine the decoupling coordinates of each sub-segment; based on the decoupling coordinates of each sub-segment, convert the deterministic steady-state response curve into an independent mapping relationship between the dynamic response coordinates and the speed and response amplitude; and establish a Kriging model based on the uncertainty parameters and the independent mapping relationship after decoupling; A generation module is used to generate a new Latin hypercube sampling sample space using Monte Carlo simulation, and to predict the mapping relationship between the dynamic response coordinates of each sample in the sample space and the speed and response amplitude through the established Kriging model, thereby generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system; The analysis module is used to analyze and obtain the uncertainty steady-state response of the three-dimensional asymmetric excitation rotor based on the steady-state response envelope curve that characterizes the uncertainty of the excitation rotor system.

[0014] The present invention provides a method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor, which has the following beneficial effects: The present invention proposes an efficient method for quantifying the uncertainty of steady-state response of an excitation rotor. By establishing a three-dimensional finite element dynamic model of a three-dimensional asymmetric excitation rotor, considering interval uncertainty, and using the dynamic response decoupling coordinate method to suppress false resonance peaks, a proxy model based on Kriging and Latin hypercube sampling methods is combined to establish a rapid analysis method for the uncertainty steady-state response of a three-dimensional asymmetric rotor system. This solves the problem of false resonance peaks that are prone to occur when the interval method is used for uncertainty analysis of the steady-state response of an asymmetric rotor. At the same time, the sample number requirement in the uncertainty quantification process is reduced, and the accuracy of the steady-state response uncertainty analysis is improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 This is a flowchart of a three-dimensional asymmetric rotor uncertainty steady-state response analysis method and verification in an embodiment of the present invention; Figure 2 This is a diagram of the excitation rotor model in an embodiment of the present invention; Figure 3 Schematic diagram of an asymmetric shaft system response test platform according to an embodiment of the present invention; Figure 4 Schematic diagram of the main part of the asymmetric shaft response test platform in an embodiment of the present invention; Figure 5: is a diagram showing the relationship between the system deterministic response amplitude and the rotation speed in an embodiment of the present invention; Figure 6 This is a graph showing the interval response-rotation speed of an isotropic system using a scanning method in an embodiment of the present invention; Figure 7 This is a graph showing the interval response-speed curve of an isotropic system using the rectangular coordinate tensor product method in an embodiment of the present invention; Figure 8 Graphs showing the system interval response and boundary error-speed curves obtained using the dynamic response decoupling coordinate tensor product method in an embodiment of the present invention. DETAILED DESCRIPTION

[0016] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0017] This paper proposes a method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor. By establishing a three-dimensional finite element dynamic model of an excitation rotor with axial slots, an asymmetric shaft system response test experimental platform is built to verify the system. Then, considering the interval uncertainty, the dynamic response decoupling coordinate method is used to suppress the false resonance peak. Combined with the proxy model based on Kriging and Latin hypercube sampling methods, a fast analysis method for the uncertainty steady-state response of the three-dimensional asymmetric rotor system is established. The influence of single and multiple uncertain parameters on the steady-state response of the three-dimensional asymmetric rotor system is studied. Figure 1 As shown, the specific steps include: S1. Establish a three-dimensional finite element dynamic model of the excitation rotor with axial slots.

[0018] The object of the present invention is an excitation rotor (such as Figure 2 ). The circumferential direction of the excitation rotor core is provided with wire slots for placing the excitation winding, and the relative angle between the wire slots is 180°. Considering that the focus of the present invention is the response of the excitation rotor bearing system, other structures have been appropriately simplified. Specifically, the actual fan structure is simplified to a disc structure, while details such as the collector ring and the excitation winding are ignored. The existence of the wire slots destroys the axial symmetry of the excitation rotor, making the rotor structure asymmetric. Therefore, the excitation rotor will experience double-frequency vibration, and a gravity resonance peak will appear at the semi-critical speed. The asymmetry of the excitation rotor will increase the number of resonance peaks, increase the probability of resonance in the rotor system, and is not conducive to the vibration safety of the rotor. In order to suppress the double-frequency vibration of the excitation rotor, it is necessary to reduce the stiffness difference of the excitation rotor in the two main inertia axes, and the compensation slots can be arranged along the axial direction, such as Figure 2 shown.

[0019] The excitation rotor is an asymmetric rotor, and its dynamic equation is: (1) Where: —excitation rotor mass matrix; —Coriolis damping matrix of the excitation rotor; —Static stiffness matrix of the excitation rotor; —Excitation rotor rotation softening matrix; —Displacement response of the two-pole generator bearing system; — velocity vector of the two-pole generator bearing system; — acceleration vector of the two-pole generator bearing system; —Gravity vector.

[0020] The bearing model of the present invention has eight coefficients, and the eight coefficients include: main stiffness coefficient and , cross stiffness coefficient and , main damping coefficient and and the cross damping coefficient and . And the anisotropy of the bearing stiffness is only reflected in the main stiffness coefficient. At the same time, the main function of the bearing damping is to suppress the response amplitude at the resonance peak, and the anisotropy of the bearing damping has little effect on the response. Therefore, the present invention ignores the anisotropy of the bearing damping. Based on the above assumptions, in a fixed coordinate system, the bearing parameters used in the present invention are:

[0021] (2) Where: ——bearing stiffness matrix in fixed coordinate system; ——Bearing damping matrix in a fixed coordinate system; In order to quantify the anisotropy of bearing stiffness, the following dimensionless bearing stiffness anisotropy coefficient is defined: β : (3) The main stiffness coefficient of the bearing can be expressed as , ,in is the average principal stiffness coefficient of the bearing, and .

[0022] The bearing matrix in the rotating coordinate system can be written as: (4) (5) Where: ——Bearing stiffness matrix in the rotating coordinate system; ——bearing damping matrix in the rotating coordinate system; ——cosine modulation term of bearing stiffness; ——Sinusoidal modulation term of bearing stiffness; ——static stiffness component of the bearing; (6) Where: ——Bearing static stiffness coefficient; ——direct bearing stiffness component; ——bearing cross stiffness component; ——bearing stiffness correction term; ——Equivalent stiffness correction caused by rotation; By assembling the rotor matrix and the bearing matrix, the overall dynamic equation of the excitation rotor system can be obtained: (7) Where: (8) in —Coriolis damping matrix of the rotor; ——Static stiffness matrix of the rotor; ——Dynamic stiffness matrix of the rotor.

[0023] S2. An asymmetric shaft system response test platform was built. The test platform consists of an asymmetric shaft system body, a drive system and a measurement system. The asymmetric shaft system body is composed of an excitation rotor. The left end of the two-pole excitation rotor is a rolling bearing, and the right end is a sliding bearing. The length of the excitation rotor is 320mm, and the diameter of the rotor main shaft is 9.5mm. The drive system consists of a speed regulator and a DC motor. The measurement system includes a DC regulated power supply, an eddy current displacement sensor, and a data collector. The steady-state response amplitude-speed relationship of the isotropic and anisotropic systems is demonstrated. In the test system, the eddy current displacement sensor is used to pick up the vibration displacement signal of the excitation rotor near the bearing. The sensors are arranged only in the horizontal direction. The data collector uses the DASH8HF-HS system, and the sampling rate is set to 20kHz. The schematic diagram of the main part of the response test platform is shown as follows. Figure 4 shown.

[0024] This paper studies the interval response of the excitation rotor under isotropic and anisotropic bearing support, which are called isotropic system and anisotropic system respectively. In the isotropic bearing condition, the parameters of bearing 1 are k av,1 =6.0×105N·m -1 , c xx,1 = c yy,1 = c=100N·s·m -1 , the parameters of bearing 2 are k av,2 =5.0×105N·m -1 , c xx,2 = c yy,2 =c=100N·s·m -1 In an isotropic system, β 1= β 2=0. In an anisotropic system, β 1= β 2=0.5. Taking into account the relative angle between the bearings γ , to quantify the bearing installation error, in anisotropic systems, the relative angle between bearings γ =45°. Figure 5 The relationship between the system's deterministic steady-state response amplitude and speed is shown. The response amplitudes studied are all responses at the center node of the left fan disk.

[0025] S3. Quantify the uncertainty of the excitation rotor system and establish a dynamic model of the excitation rotor system with interval uncertainty.

[0026] The present invention mainly takes the excitation rotor system as the object and studies the influence of interval uncertainty parameters on the steady-state response of the rotor system. In S1, the installation angles of the left and right bearings are the same. However, in actual situations, due to human installation problems, there may be differences in the installation angles between the bearings. Therefore, the present invention treats the bearing installation angle as an interval parameter, taking into account the uncertainty of the bearing installation angle. Without loss of generality, let the left bearing coincide with the XOY coordinate system, and let a certain installation angle exist between the right bearing and the XOY coordinate system. γ The bearing stiffness matrix and damping matrix on the left are consistent with formula (2), that is, , The right bearing can be considered as the left bearing rotated by an angle γ If it is formed later, then:

[0027] (9) According to the above formula, it can be found that the isotropic bearing matrix remains isotropic after rotation, so the anisotropy of the rotated bearing only exists in the bearing stiffness.

[0028] Considering the parameter uncertainty, it is assumed that there is n u interval parameters, the interval vector can be expressed as: (10) Interval vectors can also be expressed in the form of median and interval radius: (11) in: (12) Where: — interval vector; , — vectors of lower and upper bounds of the interval; — median of the interval vector; —The radius of the interval vector.

[0029] According to the dynamic equation of the deterministic excitation rotor system shown in formula (7), the dynamic equation of the excitation rotor system with interval uncertainty can be written as: (13) S4. Considering interval uncertainty, the dynamic response decoupling coordinate method is used to suppress false resonance peaks. Combined with the surrogate model based on Kriging and Latin hypercube sampling methods, a rapid analysis method for the uncertain steady-state response of a three-dimensional asymmetric rotor system is established. The influence of single and multiple uncertain parameter combinations on the steady-state response of the three-dimensional asymmetric rotor system is studied.

[0030] 1) Establish Latin hypercube sampling sample space Assume the uncertainty parameter in equation (13) to be , each parameter follows a uniform distribution , mapping the parameters to the standard hypercube space , the normalized mapping is: (14) Then divide the dimensional subspace into In each subinterval (layer), randomly select a point and randomly arrange the layer order of each dimension to avoid sample aggregation. Randomly arrange and combine the sampling points of each dimension. Combined with the sampling requirements, the number of samples is:

[0031] (15) in, is the truncation order of the surrogate model, is the uncertainty dimension. Then the sample points are moved from Mapping back to the original parameter interval:

[0032] (16) Finally, we get the Latin hypercube sampling sample point set .

[0033] 2) Perform response analysis on each sample point in the sampling sample space For each sample point , the complex exponential harmonic balance method is used to solve the steady-state response of the dynamic equation 13: Assuming the response is a periodic function, it can be expressed as a complex exponential series: (17) in, is the order of the harmonic, is the harmonic cutoff order, The complex amplitude of each harmonic component in the response, is the fundamental angular frequency of the excitation, is the complex amplitude of each order harmonic component in the excitation.

[0034] Then substitute the expanded form into the dynamic equation and match the coefficients of each harmonic term to obtain the linear equation system: (17) Finally, the frequency domain is solved by matrix inversion or iterative method to solve the domain coefficients , reconstruct the time domain response, the speed-amplitude curve of each sample is a discrete point set: (18) in is the discrete number of speed points, (vibration amplitude).

[0035] 3) Dynamic response decoupling coordinate transformation In uncertainty analysis, the phenomenon of false resonance peaks may occur. PCE research on asymmetric rotor systems shows that this phenomenon occurs in both mean and variance curves. Increasing the order of the polynomial cannot eliminate false resonance peaks, because near the resonance peak, the system response is extremely sensitive to changes in uncertain parameters and is difficult to describe with a polynomial. Related research uses polynomial dimensional decomposition technology to analyze the uncertainty response of two-degree-of-freedom spring systems and six-degree-of-freedom nonlinear rotor systems. The results show that this technology can effectively solve the dynamic response of the system, but still produces false resonance peaks at the resonance peak. This phenomenon can be weakened by increasing the order of the technology, but it cannot be completely eliminated. In order to improve this problem, the present invention uses an uncertainty quantification method for the steady-state response of a three-dimensional asymmetric rotor based on motion response decoupling coordinates.

[0036] The motion path length defined in the motion response decoupling coordinate method refers to the trajectory length of the response curve. The specific implementation steps of this method are as follows: First, mark the extreme points of the response curve and initialize the motion response decoupling coordinates of the starting point to 0. When traversing along the curve trajectory, the motion response decoupling coordinate value increases by 1 unit for each extreme point detected, and finally increases by 1 unit when reaching the end point of the curve to complete the coordinate system construction (initial construction of the speed-amplitude plane coordinates of the response curve). Secondly, the response curve is divided into several sub-segments based on the identified starting point, extreme point, and end point. For the i-th analytical point located in the j-th sub-segment, its motion response solution is expressed as:

[0037] (19) Where: —The length scale coordinate of the trajectory of the i-th solution; —The length of the starting point of the j-th segment response curve; —The total trajectory length of the j-th segment response curve. Based on the motion response decoupling coordinate (η), the single-variable mapping relationship between this coordinate and the rotational speed (Ω) and the response amplitude (A) can be constructed respectively to achieve the separate characterization of the dynamic parameters. In the reconstructed coordinate system, the multi-dimensional correlation relationship (Ω, A) in the original Cartesian coordinate system is decoupled into two independent dimensions: the trajectory length proportional coordinate-rotational speed (η, Ω) and the trajectory length proportional coordinate-amplitude (η, A). It is worth noting that the correlation relationship between the response amplitude and the random variable is established based on the unified motion response decoupling coordinate benchmark.

[0038] Regardless of whether the response solution is close to the resonance peak, the response amplitude varies relatively smoothly with the random variable, which reduces the difficulty of establishing a surrogate model. The uncertainty quantification implementation steps based on the motion response decoupling coordinate method are as follows: First, the nonlinear steady-state response curve is numerically solved for each sample point. Second, the trajectory length scale coordinates of the resulting response curve are calculated, and a bidirectional mapping relationship between the trajectory length scale coordinates and the system speed, and between the trajectory length scale coordinates and the vibration amplitude, is established. Third, using the trajectory length scale coordinates as independent variables, parameterized models for the system speed and vibration amplitude are established, respectively, and the uncertainty distribution characteristics of each are quantified using interval analysis methods. Finally, by synchronously matching the speed interval and amplitude interval parameters under the same trajectory length scale coordinates, three-dimensional data fusion and visualization reconstruction are performed in a Cartesian coordinate system to obtain the steady-state response envelope curve that represents the system uncertainty. The following is a detailed process of coordinate transformation of the steady-state response curves of each sample point in the sample space using the motion response decoupling coordinate method.

[0039] First, perform extreme point detection for each response curve. , detect the maximum and minimum points by numerical differentiation, and divide the curve into S sub-segments. Then define the length of the sub-segment trajectory and set the first Segment starting point trajectory length coordinates ,end Then map the coordinates in the sub-segment. Section No. For a point, its coordinates are:

[0040] (20) in, is the cumulative trajectory length from this point to the starting point of the sub-segment, is the total trajectory length of the sub-segment, calculated by numerical integration: (twenty one) Then decouple the corresponding mapping relationship and build and An independent mapping table is used to achieve variable decoupling.

[0041] 4) Constructing Kriging surrogate model First, define the input and output of the proxy model, where the input parameter is the uncertainty parameter ,in Represents n-dimensional real space. The output parameters are Corresponding speed and amplitude .

[0042] Assume that the output follows a Gaussian process : (twenty two) The mean function Usually a constant or linear term is taken, the covariance function Use Gaussian kernel: (twenty three) Then optimize the hyperparameters by maximum likelihood estimation ,in is the output variance, is the length scale of the nth dimension, the optimized hyperparameter for: (twenty four) in is the covariance matrix, which is represented by the covariance function calculate. is the training data, is the mean function value (such as or ).

[0043] For each trajectory length Values ​​correspond to training two Kriging models respectively: (25) (26) 5) Using surrogate models to predict uncertain response boundaries Generated using Monte Carlo simulation Latin hypercube samples , use the trained Kriging model to predict each sample and Mapping. Then the speed and amplitude are statistically analyzed. For each fixed trajectory length Values, calculate the extreme values ​​of the response for all Monte Carlo samples:

[0044] Speed ​​limit: (27) (28) Amplitude Bounds: (29) 0) Perform inverse coordinate transformation on the steady-state response boundary to obtain the steady-state response boundary in the original coordinate system Will and Restore to the original coordinate system according to the length ratio of the sub-segment trajectory (the original coordinate system refers to the steady-state response solution obtained by solving the dynamic equation using the complex exponential harmonic balance method after the first sampling, specifically the speed-amplitude plane coordinate system for the initial construction of the response curve). For each sub-segment ,according to , calculate the corresponding speed and amplitude : (30) (31) The present invention proposes a rapid analysis method for the dynamic characteristics of a three-dimensional asymmetric rotor system for the excitation rotor, and proposes an efficient method for quantifying the uncertainty of steady-state response, including verification of the asymmetric shaft system response test experimental platform, Kriging Latin hypercube sampling, dynamic response decoupling coordinate transformation, and establishment of a proxy model search boundary. It solves the problem of false resonance peaks that are prone to appear when the interval method is used for uncertainty analysis of the steady-state response of asymmetric rotors, and at the same time reduces the number of samples required in the uncertainty quantification process.

[0045] By comparing the scanning method, tensor product method, and Latin hypercube sampling method, the effectiveness of the dynamic response decoupled coordinate method proposed in this invention and the efficiency of the Latin hypercube sampling method are demonstrated. Specifically, the tensor product method can save 98.75% of the computational time compared to the scanning method in one-dimensional uncertainty problems, and the Latin hypercube sampling method can further save 44% and 99.75% of the computational time compared to the tensor product method in three-dimensional and eight-dimensional uncertainty problems, respectively. In one-dimensional and three-dimensional uncertainty problems, the relative error of the tensor product method compared to the scanning method, and the relative error of the Latin hypercube sampling method compared to the tensor product method, is basically no more than 1%, meeting the accuracy requirements of engineering analysis. At the same time, the false resonance peak phenomenon is eliminated in the above-mentioned uncertainty problems, broadening the scope of application of the interval method.

[0046] Based on the above inventive concept, the present invention proposes an embodiment to specifically illustrate and verify a method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric rotor: This example aims to verify the superiority of the motion response decoupling coordinates proposed in this invention and the efficiency and accuracy of Kriging-based Latin hypercube sampling. The sampling strategy of the scanning method is to sample the interval parameters of each dimension at equal intervals, then combine the sample points in all dimensions into a tensor product space. By performing deterministic analysis on each sample point in the space, the steady-state response of all sample points is obtained, and finally a statistical analysis is performed to obtain the response boundaries. The scanning method is a very robust method that always obtains accurate results, so it is used in this example to verify the new method.

[0047] This paper considers the elastic modulus and density of the excitation rotor, the principal stiffness coefficients, principal damping coefficients, and anisotropy parameters of the bearings, as well as the relative angle between bearings related to installation errors, as uncertain parameters. These uncertain parameters affect not only the steady-state response of the system but also its stability. This section examines seven operating conditions with various combinations of uncertain parameters. The fluctuation coefficients for all conditions are shown in Table 1.

[0048] Table 1 Uncertainty parameters under different working conditions First, the application of motion response decoupling coordinates in the analysis of uncertain steady-state response of the excitation rotor is studied using the above example. The fluctuation coefficient of the elastic modulus in this working condition is 4%. Figure 6 and Figure 7 The interval response results of the isotropic system obtained by the scanning method and the tensor product method are shown respectively. Figure 6 As shown in , the interval response results of 400 samples and 800 samples are in good agreement, indicating that the interval response results when the number of samples is 400 have converged. Figure 7As shown in (a), using the tensor product method based on rectangular coordinates, when the number of sampling points is 5, the response curve has false resonance peaks at the 3rd resonance peak, the 2nd anti-resonance peak and the 5th resonance peak, that is, these resonance peaks are not caused by parameter uncertainty, but by the polynomial proxy model incorrectly predicting the steady-state response. The resonance band only appears at the last 3 peaks. This is because the 1st and 2nd resonance peaks are not very sensitive to the elastic modulus, and there is no shift in the resonance frequency, while the 3rd, 4th and 5th resonance peaks are more sensitive to the elastic modulus, and a wider resonance band appears. However, since the amplification factor of the 4th resonance peak is small, no obvious false resonance peak appears. Similarly, the 2nd anti-resonance peak is highly sensitive to the elastic modulus, so an obvious false resonance peak also appears. From the above analysis, it can be seen that false resonance peaks tend to appear at resonance peaks with larger amplification factors. As shown in Figure 7 (b) to Figure 7 As shown in Figure (d), increasing the number of sampling points to 10, 15, and 20, respectively, reveals that the false resonance peak phenomenon improves with increasing sampling points. In particular, when the number of sampling points increases to 10, the false resonance peaks at the third and fifth resonance peaks have essentially disappeared. However, even when the number of sampling points increases to 20, the false resonance peak at the second antiresonance peak can only be improved, not completely eliminated. This is mainly because the response amplitude at this resonance peak varies dramatically, making it difficult to describe using a polynomial.

[0049] Figure 8 The isotropic system interval response and its boundary error-speed curve obtained by the tensor product method based on the motion response decoupling coordinates are shown. It can be found that the application of the motion response decoupling coordinates eliminates the occurrence of false resonance peaks at each resonance peak and antiresonance peak. The results obtained by the tensor product method based on the motion response decoupling coordinates are similar to those of the scanning method ( Figure 6 ) are very consistent, with prediction errors generally no higher than 1%. In terms of efficiency, the tensor product method based on motion response decoupled coordinates uses 5 samples, while the scanning method uses 400 samples. This method can save 98.75% of computation time, improving computational efficiency.

[0050] Based on the same inventive concept, the present invention also proposes a three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis system, comprising: The dynamic model construction module is used to establish the dynamic model of the excitation rotor system containing interval uncertainty by introducing interval uncertainty parameters based on the three-dimensional finite element dynamic model of the excitation rotor.

[0051] A solution module is used to map the uncertain parameters in the dynamic model of the excitation rotor system to the standard hypercube space using the Latin hypercube sampling method, generate a Latin hypercube sampling sample space, perform response analysis on each sample point in the sample space, and establish a deterministic steady-state response curve for each sample point; perform extreme value detection on the deterministic steady-state response curve, and divide the deterministic steady-state response curve into multiple sub-segments according to the detected extreme value points; decouple the speed and response amplitude under the dynamic response coordinates and determine the decoupling coordinates of each sub-segment; based on the decoupling coordinates of each sub-segment, convert the deterministic steady-state response curve into an independent mapping relationship between the dynamic response coordinates and the speed and response amplitude; and establish a Kriging model based on the uncertainty parameters and the independent mapping relationship after decoupling.

[0052] A generation module is used to generate a new Latin hypercube sampling sample space using Monte Carlo simulation, and to predict the mapping relationship between the dynamic response coordinates of each sample in the sample space and the speed and response amplitude through the established Kriging model, thereby generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system; The analysis module is used to analyze and obtain the uncertainty steady-state response of the three-dimensional asymmetric excitation rotor based on the steady-state response envelope curve that characterizes the uncertainty of the excitation rotor system.

[0053] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

Claims

1. A method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor, characterized in that: The following steps are involved: Based on the three-dimensional finite element dynamic model of the three-dimensional asymmetric excitation rotor, the dynamic model of the excitation rotor system with interval uncertainty is established by introducing interval uncertainty parameters. The Latin hypercube sampling method is used to map the uncertain parameters in the dynamic model of the excitation rotor system to the standard hypercube space to generate a Latin hypercube sampling sample space. Response analysis is performed on each sample point in the sample space, and a deterministic steady-state response curve is established for each sample point. Extreme value detection is performed on the deterministic steady-state response curve, and the deterministic steady-state response curve is divided into multiple sub-segments based on the detected extreme value points. The speed and response amplitude are decoupled in the dynamic response coordinate to determine the decoupling coordinates of each sub-segment. Based on the decoupling coordinates of each sub-segment, the deterministic steady-state response curve is converted into an independent mapping relationship between the dynamic response coordinates and the speed and response amplitude. A Kriging model is established based on the uncertainty parameters and the independent mapping relationship after decoupling. Monte Carlo simulation is used to generate a new Latin hypercube sampling space. The mapping relationship between the dynamic response coordinates of each sample in the sample space and the speed and response amplitude is predicted through the established Kriging model, generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system. Based on the steady-state response envelope curve that characterizes the uncertainty of the excitation rotor system, the uncertainty steady-state response of the three-dimensional asymmetric excitation rotor is analyzed and obtained.

2. The method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1 is characterized in that: The three-dimensional finite element dynamic model based on the three-dimensional asymmetric excitation rotor introduces interval uncertainty parameters to establish an excitation rotor system dynamic model containing interval uncertainty, which specifically includes the following steps: Establish a three-dimensional finite element dynamic model of a three-dimensional asymmetric excitation rotor: ； Definition of the existence of excitation rotor system n u interval uncertainty parameters, the interval vector is expressed as: ； Represent an interval vector as a median and interval radius: ； in: ； Where: is an interval vector; 、 are the lower and upper bound vectors of the interval; is the median of the interval vector; is the radius of the interval vector; According to the three-dimensional finite element dynamic model of the excitation rotor, the dynamic equation of the excitation rotor system with interval uncertainty is obtained as follows: ； in, is the mass matrix of the excitation rotor; is the Coriolis damping matrix of the excitation rotor; is the static stiffness matrix of the excitation rotor; Softening matrix for excitation rotor rotation; is the displacement response of the two-pole generator bearing system; is the velocity vector of the two-pole generator bearing system; is the acceleration vector of the two-pole generator bearing system; is the gravity vector; is the cosine modulation term of the bearing stiffness; is the sinusoidal modulation term of the bearing stiffness; , is the Coriolis damping matrix of the rotor, is the static stiffness matrix of the rotor, is the dynamic stiffness matrix of the rotor, is the static stiffness component of the bearing, is the fundamental angular frequency of the excitation.

3. The method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1 is characterized in that: The step of performing extreme value detection on the deterministic steady-state response curve and dividing the deterministic steady-state response curve into a plurality of sub-segments according to the detected extreme value points specifically includes the following steps: Mark the extreme points of the deterministic steady-state response curve and initialize the motion response decoupling coordinates of the starting point to 0; When traversing the trajectory of the deterministic steady-state response curve, the motion response decoupling coordinate value increases by 1 unit each time an extreme point is detected, and increases by 1 unit again when reaching the end point of the deterministic steady-state response curve; The deterministic steady-state response curve is divided into several sub-segments based on the identified starting points, extreme points, and end points.

4. The method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1 is characterized in that: The Latin hypercube sampling method is used to map the uncertain parameters in the dynamic model of the excitation rotor system to the standard hypercube space to generate the Latin hypercube sampling sample space; specifically, the following steps are included: The uncertainty parameter in the dynamic equation of the excitation rotor system with interval uncertainty is set to , each parameter follows a uniform distribution , mapping the parameters to the standard hypercube space , the normalized mapping is: ； in, a i 、 b i Refers to each uncertainty parameter Upper and lower limits; Divide the dimensional subspace into In each subinterval, a sample point is randomly selected and the order of intervals in each dimension is randomly arranged. The sample points of each dimension are randomly arranged and combined, and the number of samples is: ； in, is the truncation order of the surrogate model, is the uncertainty dimension; The sample points from Mapping back to the original parameter interval, we get: ； Finally, the Latin hypercube sampling sample point set is .

5. The method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor according to claim 4 is characterized in that: The step of performing response analysis on each sample point in the sample space and establishing a deterministic steady-state response curve for each sample point specifically includes the following steps: For each sample point , the complex exponential harmonic balance method is used to solve the steady-state response of the dynamic equation of the excitation rotor system with interval uncertainty: ； Assuming the steady-state response to be a periodic function, it can be expressed using a complex exponential series: ； in, is the order of the harmonic, is the harmonic cutoff order, is the complex amplitude of each order harmonic component in the response, is the fundamental angular frequency of the excitation, is the complex amplitude of each order harmonic component in the excitation; is the displacement response vector of the generator bearing system with respect to time; About time t The gravity vector of Substituting the expanded form of the steady-state response into the dynamic equation and matching the coefficients of each harmonic term, we obtain the linear equation system: ； in, is the mass matrix, that is, the harmonic order The inertia term, is the motion response solution; Solve for domain coefficients by matrix inversion or iterative methods , reconstruct the time domain response, and obtain the deterministic steady-state response curve of each sample point: ； in is the number of discrete speed points, is the speed value of each sample point of the response curve, The response amplitude of each sample point is .

6. The method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor according to claim 3 is characterized in that: The method of establishing a Kriging model based on the uncertainty parameters and the independent mapping relationship after decoupling specifically includes the following steps: Define input parameters as uncertainty parameters ,in express n dimensional real space, the output parameter is the length of each sub-segment trajectory Corresponding speed and amplitude ; Set the output parameter to follow the Gaussian process : ； in, is the covariance function that controls the speed, is the covariance function that controls the response amplitude, the mean function Take a constant or linear term, covariance function Using the Gaussian kernel, we get: ； Optimizing hyperparameters via maximum likelihood estimation ,in is the output variance, For the n dimensional length scale, optimized hyperparameters for: ； in is the covariance matrix, which is represented by the covariance function calculate; is the training data, 、 、 is the mean function value; For each trajectory length , respectively train the Kriging model: ； 。 7. The method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1 is characterized in that: The Monte Carlo simulation is used to generate a new Latin hypercube sampling sample space, and the mapping relationship between the dynamic response coordinates of each sample in the sample space and the speed and response amplitude is predicted by the established Kriging model to generate a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system; The specific steps include: Generate new Latin hypercube sample sets using Monte Carlo simulation; The established Kriging model is used to predict the mapping relationship between the deterministic steady-state response curve trajectory length of each sample and the speed and response amplitude, and the speed and amplitude boundaries of each sample are determined. The speed and amplitude boundaries are mapped back to the original deterministic steady-state response curve according to the sub-segment trajectory length, and a steady-state response envelope curve is generated that can characterize the uncertainty of the excitation rotor system.

8. A three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis system, characterized in that: include: The dynamic model construction module is used to establish the dynamic model of the excitation rotor system with interval uncertainty by introducing interval uncertainty parameters based on the three-dimensional finite element dynamic model of the three-dimensional asymmetric excitation rotor; A solution module is used to map the uncertain parameters in the dynamic model of the excitation rotor system to the standard hypercube space using the Latin hypercube sampling method, generate a Latin hypercube sampling sample space, perform response analysis on each sample point in the sample space, and establish a deterministic steady-state response curve for each sample point; perform extreme value detection on the deterministic steady-state response curve, and divide the deterministic steady-state response curve into multiple sub-segments based on the detected extreme value points; decouple the speed and response amplitude under the dynamic response coordinate to determine the decoupling coordinates of each sub-segment; based on the decoupling coordinates of each sub-segment, convert the deterministic steady-state response curve into an independent mapping relationship between the dynamic response coordinates and the speed and response amplitude; and establish a Kriging model based on the uncertainty parameters and the independent mapping relationship after decoupling; A generation module is used to generate a new Latin hypercube sampling sample space using Monte Carlo simulation, and to predict the mapping relationship between the dynamic response coordinates of each sample in the sample space and the speed and response amplitude through the established Kriging model, thereby generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system; The analysis module is used to analyze and obtain the uncertainty steady-state response of the three-dimensional asymmetric excitation rotor based on the steady-state response envelope curve that characterizes the uncertainty of the excitation rotor system.

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