Three-dimensional unsymmetrical field-rotor uncertainty steady-state response analysis method and system

Abstract

The application discloses a three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis method and system, relates to the energy power technology field, and comprises the following steps: a three-dimensional finite element dynamics model of an excitation rotor with an axial slot is established; uncertainty quantification is carried out on the excitation rotor system, and a dynamics model of the excitation rotor system containing interval uncertainty is established; a Latin hypercube sampling method is adopted, and a Kriging surrogate model is used to solve the uncertainty steady-state response; considering the interval uncertainty, a dynamic response decoupling coordinate method is used to suppress false resonance peaks; a fast analysis method of the three-dimensional asymmetric rotor system uncertainty steady-state response is established by combining the Kriging and Latin hypercube sampling method-based surrogate model; and the three-dimensional asymmetric rotor system steady-state response is analyzed; and the method solves the phenomenon that the interval method is prone to false resonance peaks when used for uncertainty analysis of the asymmetric rotor steady-state response.

Description

Technical Field

[0001] This invention relates to the field of energy and power technology, specifically to a method and system for analyzing the uncertainty steady-state response of a three-dimensional asymmetric excitation rotor. Background Technology

[0002] To meet power generation demands, the excitation rotor in a steam turbine unit is slotted along its axial direction, making its cross-section non-circular. This results in different moments of inertia along the two principal axes of inertia, leading to differences in bending stiffness, i.e., axial asymmetry. Typically, transverse circular arc slots or axial slots are added to the large teeth in the design to reduce this asymmetry and lower vibration. In early designs, trial-and-error methods were often used to determine the slotting method, but a systematic and effective approach has not yet been developed. 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 geometric model parameters of the rotor and boundary condition parameters such as bearing supports all have definite values. Even when studying the influence of these parameters on dynamic characteristics, the parameters are typically set to a finite number of parameters, and the relationship between the dynamic characteristics and these parameters is investigated. All of these studies can be termed deterministic research.

[0004] However, uncertainty is widespread in reality. When multiple parameters are uncertain simultaneously, it will have a significant impact on the rotor dynamics. Therefore, it is necessary to conduct research on the quantification of rotor system uncertainties. Monte Carlo and scanning methods are two robust and convenient computational methods, but their drawbacks include the need for a large number of samples to ensure computational convergence. Furthermore, the number of samples increases exponentially with the dimensionality of the uncertainty parameters, leading to the "curse of dimensionality" in high-dimensional cases, making computation difficult. Moreover, previous research on rotor system uncertainty quantification has focused on simple models such as beam elements, with few successful applications to three-dimensional rotor systems, especially asymmetric rotors. To overcome this problem, polynomial chaotic expansion (PCE) and polynomial interval methods have been developed to establish surrogate models of stochastic processes. However, these two methods require few samples, are computationally fast, and are prone to generating spurious resonance peaks, leading to errors in the analysis results. Summary of the Invention

[0005] To address the shortcomings of existing technologies in quantifying the uncertainty of nonlinear steady-state responses of large asymmetric rotors—namely, difficulties in convergence, long processing times, and the tendency to produce spurious resonance peaks leading to erroneous analysis results—this 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 uncertainties, and using the dynamic response decoupling coordinate method to suppress spurious resonance peaks, combined with a surrogate model based on Kriging and Latin hypercube sampling methods, a rapid analysis method for the uncertainty steady-state response of a three-dimensional asymmetric rotor system is established, thus solving the problems existing in the prior art.

[0006] A method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor includes the following steps:

[0007] Based on the three-dimensional asymmetric excitation rotor, a three-dimensional finite element dynamic model is established by introducing interval uncertainty parameters to create a dynamic model of the excitation rotor system containing interval uncertainty.

[0008] The Latin hypercube sampling method is used to map the uncertain parameters in the dynamic model of the excitation rotor system to a standard hypercube space, generating a Latin hypercube sampling sample space. Response analysis is performed on each sample point in the sample space to establish a deterministic steady-state response curve for each sample point. Extreme value detection is performed on the deterministic steady-state response curve, and the curve is divided into multiple sub-segments based on the detected extreme points. The rotational speed and response amplitude are decoupled in the dynamic response coordinate system, and the decoupling coordinates for each sub-segment are determined. Based on the decoupling coordinates of each sub-segment, the deterministic steady-state response curve is transformed into an independent mapping relationship between the dynamic response coordinates and the rotational speed and response amplitude. Based on the uncertain parameters and the decoupled independent mapping relationship, a Kriging model is established.

[0009] A new Latin hypercube sampling sample space is generated using Monte Carlo simulation. The mapping relationship between the dynamic response coordinates of each sample in the sample space and the rotational speed and response amplitude is predicted by the established Kriging model, thereby generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system.

[0010] Based on the steady-state response envelope curve characterizing the uncertainty of the excitation rotor system, the uncertain steady-state response of the three-dimensional asymmetric excitation rotor is analyzed.

[0011] Furthermore, the three-dimensional finite element dynamic model based on the three-dimensional asymmetric excitation rotor, by introducing interval uncertainty parameters, establishes a dynamic model of the excitation rotor system containing interval uncertainty, specifically including the following steps:

[0012] Establish a three-dimensional finite element dynamic model of a three-dimensional asymmetric excitation rotor:

[0013] ;

[0014] Define the presence of excitation rotor system n u If there are interval uncertainty parameters, then the interval vector is represented as:

[0015] ;

[0016] Express the interval vector in the form of the midpoint and the interval radius:

[0017] ;

[0018] in:

[0019] ;

[0020] In the formula: It is an interval vector; , Let these be the lower and upper bound vectors of the interval; It is the midpoint of the interval vector; Let be the radius of the interval vector;

[0021] Based on the three-dimensional finite element dynamic model of the excitation rotor, the dynamic equations of the excitation rotor system containing interval uncertainties are obtained as follows:

[0022] ;

[0023] in, Here is the mass matrix of the excitation rotor; The Coriolis damping matrix of the excitation rotor; This represents the static stiffness matrix of the excitation rotor; For the rotating softening matrix of the excitation rotor; For the displacement response of the bearing system of a two-pole generator; The velocity vector of the two-pole generator bearing system; The acceleration vector of the two-pole generator bearing system; It is the gravity vector; This is the cosine modulation term for bearing stiffness; This is the sinusoidal modulation term for bearing stiffness; , Here is the Coriolis damping matrix of the rotor. Let be the static stiffness matrix of the rotor. Here is the dynamic stiffness matrix of the rotor. This represents the static stiffness component of the bearing. The fundamental angular frequency of the excitation.

[0024] Furthermore, the step of performing extreme value detection on the deterministic steady-state response curve and dividing the deterministic steady-state response curve into multiple sub-segments based on the detected extreme points specifically includes the following steps:

[0025] The extreme points of the deterministic steady-state response curve are marked, and the decoupled coordinates of the motion response at the starting point are initialized to 0.

[0026] When traversing along the trajectory of the deterministic steady-state response curve, the value of the motion response decoupling coordinate is increased by 1 unit for each extreme point detected, until the endpoint of the deterministic steady-state response curve is reached, and then increased by 1 unit again.

[0027] The deterministic steady-state response curve is divided into several segments based on the identified starting point, extreme point, and ending point.

[0028] Furthermore, 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, generating a Latin hypercube sampling sample space; specifically, this includes the following steps:

[0029] Let the uncertainty parameter in the dynamic equation of the excitation rotor system containing interval uncertainty be: Each parameter follows a uniform distribution Mapping parameters to standard hypercube space The normalized mapping is:

[0030] ;

[0031] in, a i , b i This refers to each uncertainty parameter The upper and lower limits;

[0032] Divide the dimensional subspace, dividing each dimension into... Each sub-interval has equal width. Within each sub-interval, a sample point is randomly selected and the interval order of each dimension is randomly arranged.

[0033] The sample points of each dimension are randomly arranged and combined, and the number of samples is:

[0034] ;

[0035] in, The order of the surrogate model is truncated. As an dimension of uncertainty;

[0036] Transfer sample points from Mapping back to the original parameter range, we get:

[0037] ;

[0038] The final Latin hypercube sampling point set is as follows .

[0039] 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:

[0040] For each sample point The steady-state response of the excitation rotor system dynamics equations containing interval uncertainties is solved using the complex exponential harmonic balance method:

[0041] ;

[0042] Assuming the steady-state response is a periodic function, it can be represented using a complex exponential series as follows:

[0043] ;

[0044] in, The order of the harmonic is denoted by . The harmonic cutoff order is... To respond to the complex amplitude of each harmonic component in the middle, For the fundamental angular frequency of the excitation, To determine the complex amplitude of each harmonic component in the excitation; Let be the displacement response vector of the generator bearing system with respect to time; Regarding time t The gravity vector;

[0045] Substituting the expansion of the steady-state response into the dynamic equations and matching the coefficients of each harmonic term, we obtain a system of linear equations:

[0046] ;

[0047] in, The mass matrix, which is the harmonic order. inertial term, For the solution of motion response;

[0048] The domain coefficients can be solved by matrix inversion or iterative methods. Reconstructing the time-domain response yields the deterministic steady-state response curve for each sample point:

[0049] ;

[0050] in The number of discrete points for rotational speed. For each sample point of the response curve, the rotational speed value. The response amplitude for each sample point, the vibration amplitude is .

[0051] Furthermore, the Kriging model is established based on the uncertainty parameters and the decoupled independent mapping relationship; specifically, this includes the following steps:

[0052] Define the input parameters as uncertain parameters. ,in express n The output parameter is the length of each sub-segment trajectory in a 3D real space. corresponding speed and amplitude ;

[0053] The output parameter is set to follow a Gaussian process. :

[0054] ;

[0055] in, The covariance function for controlling the rotational speed. The covariance function and mean function are used to control the response amplitude. Take a constant or linear term, covariance function Using a Gaussian kernel, we get:

[0056] ;

[0057] Optimize hyperparameters using maximum likelihood estimation. ,in To output the variance, For the first n Dimension length scale, optimized hyperparameters for:

[0058] ;

[0059] in The covariance matrix is ​​derived from the covariance function. calculate; For training data, , , The value is the mean function value;

[0060] For each trajectory length The Kriging models were trained accordingly:

[0061] ;

[0062] .

[0063] Furthermore, the method of generating a new Latin hypercube sampling sample space using Monte Carlo simulation, and predicting the mapping relationship between the dynamic response coordinates, rotational speed, and response amplitude of each sample in the sample space using the established Kriging model, generates a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system; specifically, it includes the following steps:

[0064] A new Latin hypercube sampling set was generated using Monte Carlo simulation.

[0065] By establishing the Kriging model, the mapping relationship between the trajectory length of the deterministic steady-state response curve of each sample and the rotational speed and response amplitude is predicted, and the rotational speed and amplitude boundaries of each sample are determined.

[0066] Based on the segment trajectory length, the boundaries of rotational speed and amplitude are mapped back to the original deterministic steady-state response curve, generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system.

[0067] The present invention also includes a three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis system, comprising:

[0068] The dynamic model construction module is used to build a three-dimensional finite element dynamic model based on a three-dimensional asymmetric excitation rotor. By introducing interval uncertainty parameters, a dynamic model of the excitation rotor system containing interval uncertainty is established.

[0069] The solution module is used to map uncertain parameters in the dynamic model of the excitation rotor system to a standard hypercube space using the Latin hypercube sampling method, generating a Latin hypercube sampling sample space. Response analysis is performed on each sample point in the sample space to establish a deterministic steady-state response curve for each sample point. Extrema detection is performed on the deterministic steady-state response curve, and the curve is divided into multiple sub-segments based on the detected extrema. Rotational speed and response amplitude are decoupled in dynamic response coordinates, and the decoupling coordinates for each sub-segment are determined. 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 rotational speed and response amplitude. Based on the uncertain parameters and the decoupled independent mapping relationship, a Kriging model is established.

[0070] The generation module is used to generate a new Latin hypercube sampling sample space using Monte Carlo simulation. By establishing the Kriging model, it predicts the mapping relationship between the dynamic response coordinates of each sample in the sample space and the rotational speed and response amplitude, and generates a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system.

[0071] The analysis module is used to analyze the uncertain 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.

[0072] This invention provides a method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor, which has the following advantages:

[0073] This invention proposes an efficient method for quantifying the uncertainty of steady-state response of excitation rotors. By establishing a three-dimensional asymmetric finite element dynamic model of the excitation rotor, considering interval uncertainties, and using the dynamic response decoupling coordinate method to suppress spurious resonance peaks, a rapid analysis method for the uncertain steady-state response of a three-dimensional asymmetric rotor system is established by combining a surrogate model based on Kriging and Latin hypercube sampling methods. This solves the problem of spurious resonance peaks that easily occur when using interval methods for the uncertainty analysis of the steady-state response of asymmetric rotors, while reducing the number of samples required in the uncertainty quantification process and improving the accuracy of the steady-state response uncertainty analysis. Attached Figure Description

[0074] Figure 1 This is a flowchart illustrating a three-dimensional asymmetric rotor uncertainty steady-state response analysis method and its verification in an embodiment of the present invention;

[0075] Figure 2 This is a model diagram of the excitation rotor in an embodiment of the present invention;

[0076] Figure 3 This is a schematic diagram of the asymmetric shaft system response testing experimental platform in an embodiment of the present invention;

[0077] Figure 4 This is a schematic diagram of the main body of the asymmetric shaft system response testing experimental platform in an embodiment of the present invention;

[0078] Figure 5 This is a graph showing the amplitude-speed relationship of the deterministic response of the system in an embodiment of the present invention.

[0079] Figure 6 This is a diagram showing the interval response-speed curve of an isotropic system using the scanning method in an embodiment of the present invention.

[0080] Figure 7 This is a diagram showing the interval response-rotation speed curve of an isotropic system using the rectangular coordinate tensor product method in an embodiment of the present invention.

[0081] Figure 8 The figure shows the system interval response and boundary error-speed curves obtained using the dynamic response decoupling coordinate tensor product method in this embodiment of the invention. Detailed Implementation

[0082] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0083] This invention proposes a method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor. A three-dimensional finite element dynamic model of the excitation rotor with axial slots is established, and an experimental platform for testing the response of the asymmetric shaft system is built to verify the system. Then, considering interval uncertainties, a dynamic response decoupling coordinate method is used to suppress spurious resonance peaks. Combined with a surrogate model based on Kriging and Latin hypercube sampling methods, a rapid analysis method for the uncertain 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:

[0084] S1. Establish a three-dimensional finite element dynamic model of the excitation rotor with axial slots.

[0085] The object of this invention is an excitation rotor (such as...) Figure 2 The excitation rotor core has slots along its circumferential direction for placing the excitation windings, with a relative angle of 180° between the slots. Considering that the focus of this invention is the response of the excitation rotor bearing system, other structures have been appropriately simplified. Specifically, the actual fan structure has been simplified to a disc structure, while details such as the slip rings and excitation windings have been ignored. The presence of the slots disrupts the axisymmetry of the excitation rotor, resulting in an asymmetry in the rotor structure. Therefore, the excitation rotor will exhibit second-harmonic vibration, with a gravity resonance peak appearing at the semi-critical speed. The asymmetry of the excitation rotor increases the number of resonance peaks, raising the probability of resonance in the rotor system and negatively impacting the rotor's vibration safety. To suppress the second-harmonic vibration of the excitation rotor, it is necessary to reduce the stiffness difference between the two principal inertial axes. Compensation slots can be arranged along the axial direction, such as... Figure 2 As shown.

[0086] The excitation rotor is an asymmetric rotor, and its dynamic equation is:

[0087] (1)

[0088] In the formula: —Excitation rotor mass matrix; —The Coriolis damping matrix of the excitation rotor; —The static stiffness matrix of the excitation rotor; —Excitation rotor rotation softening matrix; —Displacement response of a 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.

[0089] The bearing model of this invention is an eight-coefficient bearing model, the eight coefficients including: principal stiffness coefficients. and Cross stiffness coefficient and Principal damping coefficient and and cross-damping coefficient and Furthermore, the anisotropy of bearing stiffness is assumed to be limited to the principal stiffness coefficients. Simultaneously, the primary function of bearing damping is to suppress the response amplitude at the resonance peak, and the anisotropy of bearing damping has a relatively small impact on the response. Therefore, this invention ignores the anisotropy of bearing damping. Based on the above assumptions, the bearing parameters used in this invention under a fixed coordinate system are:

[0090] (2)

[0091] In the formula: —Bearing stiffness matrix in a fixed coordinate system; —Bearing damping matrix in a fixed coordinate system; To quantify the anisotropy of bearing stiffness, the following dimensionless bearing stiffness anisotropy coefficient is defined. β :

[0092] (3)

[0093] The principal stiffness coefficient of the bearing can then be expressed as: , ,in Let be the bearing's average principal stiffness coefficient, and .

[0094] The bearing matrix in the rotating coordinate system can be written as:

[0095] (4)

[0096] (5)

[0097] In the formula: —Bearing stiffness matrix in rotating coordinate system; —Bearing damping matrix in rotating coordinate system; —The cosine modulation term of bearing stiffness; —The sinusoidal modulation term of bearing stiffness; —The static stiffness component of the bearing;

[0098] (6)

[0099] In the formula: — Bearing static stiffness coefficient; —Direct stiffness component of the bearing; —Bearing cross stiffness component; —Bearing stiffness correction term; —Equivalent stiffness correction due to rotation; by assembling the rotor matrix and bearing matrix, the overall dynamic equation of the excitation rotor system can be obtained:

[0100] (7)

[0101] In the formula:

[0102] (8)

[0103] in —The Coriolis damping matrix of the rotor; —The static stiffness matrix of the rotor; —The dynamic stiffness matrix of the rotor.

[0104] S2. An experimental platform for testing the response of an asymmetric shaft system was constructed. The platform consists of the asymmetric shaft system body, a drive system, and a measurement system. The asymmetric shaft system body is composed of an excitation rotor, with a rolling bearing at the left end and a sliding bearing at the right end of the two-pole excitation rotor. The length of the excitation rotor is 320 mm, and the diameter of the rotor shaft is 9.5 mm. The drive system consists of a speed controller and a DC motor. The measurement system includes a DC regulated power supply, eddy current displacement sensors, and a data acquisition unit. The steady-state response amplitude-speed relationship of isotropic and anisotropic systems was demonstrated. In the test system, the eddy current displacement sensors are used to pick up the vibration displacement signal near the bearing of the excitation rotor. Sensors are only arranged in the horizontal direction. The data acquisition unit uses a DASH8HF-HS system with a sampling rate set to 20 kHz. A schematic diagram of the main body of the response testing platform is shown below. Figure 4 As shown.

[0105] This invention simultaneously studies the range response of the excitation rotor under isotropic and anisotropic bearing support, referred to as the isotropic system and the anisotropic system, respectively. In the isotropic bearing condition, the parameters of bearing 1 are: k av,1 =6.0×10⁵ N·m -1 , c xx,1 = c yy,1 = c =100 N·s·m -1 The parameters of bearing 2 are: k av,2 =5.0×10⁵ N·m -1 , c xx,2 = c yy,2 =c=100N·s·m-1 In isotropic systems, β 1= β 2 = 0. In anisotropic systems, β 1= β 2 = 0.5. The relative angle between the bearings is taken into account. γ To quantify bearing installation errors, in anisotropic systems, the relative angles between bearings γ =45°. Figure 5 The deterministic steady-state response amplitude-speed relationship of the system is shown. The response amplitudes studied are all at the center node of the left fan disk.

[0106] S3. Quantify the uncertainty of the excitation rotor system and establish a dynamic model of the excitation rotor system containing interval uncertainty.

[0107] This invention focuses on an excitation rotor system, investigating the influence of interval uncertain 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 reality, due to human error during installation, the installation angles between the bearings may differ. Therefore, this invention treats the bearing installation angle as an interval parameter, considering the uncertainty of the bearing installation angle. Without loss of generality, the left bearing is assumed to coincide with the XOY coordinate system, while the right bearing is assumed to have a certain installation angle with the XOY coordinate system. γ The stiffness matrix and damping matrix of the left bearing are consistent with equation (2), that is... , The right-side bearing can be considered as being rotated by the left-side bearing. γ If it is formed later, then:

[0108] (9)

[0109] According to the above formula, the isotropic bearing matrix remains isotropic after rotation. Therefore, the anisotropy of the rotated bearing only exists in the bearing stiffness.

[0110] Considering parameter uncertainties, assume that there are uncertainties in the excitation rotor system. n u If there are interval parameters, then the interval vector can be represented as:

[0111] (10)

[0112] An interval vector can also be represented as the midpoint and the radius of the interval:

[0113] (11)

[0114] in:

[0115] (12)

[0116] In the formula: —Interval vector; , —The lower and upper bound vectors of the interval; —The midpoint of the interval vector; — The radius of the interval vector.

[0117] Based on the dynamic equation of the deterministic excitation rotor system shown in equation (7), the dynamic equation of the excitation rotor system containing interval uncertainties can be written:

[0118] (13)

[0119] S4. Considering interval uncertainty, the dynamic response decoupling coordinate method is used to suppress false resonance peaks. Combined with a surrogate model based on Kriging and Latin hypercube sampling methods, a fast analysis method for the uncertain steady-state response of a 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.

[0120] 1) Establishing the Latin hypercube sampling sample space

[0121] The uncertainty parameter in equation (13) is set as follows: Each parameter follows a uniform distribution Mapping parameters to standard hypercube space The normalized mapping is:

[0122] (14)

[0123] Next, the dimensional subspace is divided, with each dimension divided into... Divide the sample into equal-width intervals (layers). Within each sub-interval, randomly select a point and randomly arrange the layer order for each dimension to avoid sample clustering. Randomly arrange and combine the sampling points for each dimension. Based on sampling requirements, the sample size is:

[0124] (15)

[0125] in, The order of the surrogate model is truncated. This represents the dimension of uncertainty. Then, the sample points are... Map back to the original parameter range:

[0126] (16)

[0127] Finally, the Latin hypercube sampling point set was obtained. .

[0128] 2) Perform response analysis on each sample point in the sample space.

[0129] For each sample point The steady-state response of dynamic equation 13 is solved using the complex exponential harmonic equilibrium method:

[0130] Assuming the response is a periodic function, it can be represented by a complex exponential series:

[0131] (17)

[0132] in, The order of the harmonics. The harmonic cutoff order is... The complex amplitudes of each harmonic component in the response. For the fundamental angular frequency of the excitation, The complex amplitude of each harmonic component in the excitation is given.

[0133] Then, substituting the expansion into the dynamic equations and matching the coefficients of each harmonic term, we obtain a system of linear equations:

[0134] (17)

[0135] Finally, the frequency domain is solved by matrix inversion or iterative methods to obtain the domain coefficients. The time-domain response is reconstructed, and the rotational speed-amplitude curve of each sample is a discrete set of points:

[0136] (18)

[0137] in The number of discrete points for rotational speed. (Vibration amplitude).

[0138] 3) Dynamic response decoupling coordinate transformation

[0139] In uncertainty analysis, spurious resonance peaks can occur. PCE studies of asymmetric rotor systems show this phenomenon in both mean and variance curves. Increasing the polynomial order cannot eliminate spurious resonance peaks because the system response near these peaks is extremely sensitive to changes in uncertain parameters, making it difficult to describe using a polynomial. Related studies have used polynomial dimensionality decomposition techniques to analyze the uncertainty responses of two-degree-of-freedom spring systems and six-degree-of-freedom nonlinear rotor systems. Results show that this technique can effectively solve the system's dynamic response, but it still generates spurious resonance peaks at the resonance points. Increasing the order of the technique can mitigate this phenomenon, but it cannot completely eliminate it. To improve this problem, this invention uses an uncertainty quantification method for the steady-state response of a three-dimensional asymmetric rotor based on decoupled coordinates of the motion response.

[0140] In the motion response decoupling coordinate method, the motion path length is defined as the trajectory length of the response curve. The specific implementation steps of this method are as follows: First, the extreme points of the response curve are marked, and the motion response decoupling coordinates of the starting point are initialized to 0. While traversing the curve trajectory, the motion response decoupling coordinate value is incremented by 1 unit for each detected extreme point, and then incremented again by 1 unit when the curve reaches its endpoint, completing the coordinate system construction (initially constructing the rotational speed-amplitude plane coordinates of the response curve). Second, based on the identified starting point, extreme points, and endpoint, the response curve is divided into several sub-segments. For the i-th analytical point located within the j-th sub-segment, its motion response solution is expressed as:

[0141] (19)

[0142] In the formula: —The proportional coordinate of the trajectory length of the i-th solution; —The length of the starting point of the j-th segment of the response curve; —The total trajectory length of the j-th segment of the response curve. Based on the motion response decoupling coordinates (η), univariate mapping relationships between these coordinates and rotational speed (Ω) and response amplitude (A) can be constructed, achieving separate characterization of dynamic parameters. In the reconstructed coordinate system, the multidimensional correlation (Ω,A) in the original Cartesian coordinate system is decoupled into two independent dimensions: trajectory length proportional coordinates-rotational speed (η,Ω) and trajectory length proportional coordinates-amplitude (η,A). It is worth noting that the correlation between the response amplitude and random variables is established based on a unified motion response decoupling coordinate benchmark.

[0143] Regardless of whether the response solution is close to the resonance peak, the relationship between the response amplitude and the random variable is relatively smooth, which can reduce the difficulty of establishing a surrogate model. The uncertainty quantification based on the motion response decoupling coordinate method is implemented as follows: First, the nonlinear steady-state response curves of each sample point are numerically solved; second, the trajectory length ratio coordinates of the obtained response curves are calculated, and a two-way mapping relationship is constructed between the trajectory length ratio coordinates and the system rotational speed, and between the trajectory length ratio coordinates and the vibration amplitude; third, with the trajectory length ratio coordinates as independent variables, parameterized models of the system rotational speed and vibration amplitude are established respectively, and the uncertainty distribution characteristics of the two are quantified by the interval analysis method; finally, by synchronously matching the rotational speed interval and amplitude interval parameters under the same trajectory length ratio coordinates, three-dimensional data fusion and visualization reconstruction are performed in the Cartesian coordinate system to obtain the steady-state response envelope curve characterizing the system uncertainty. The following is a detailed process of using the motion response decoupling coordinate method to perform coordinate transformation on the steady-state response curves of each sample point in the sampled sample space.

[0144] First, extreme point detection is performed on each response curve. By detecting maxima and minima using numerical differentiation, the curve is divided into S sub-segments. Then, the length of each sub-segment is defined, and the length of the segment is set. Segment starting point trajectory length coordinates ,end Then map the coordinates within the sub-segment, for the first... Section No. For each point, its coordinates are:

[0145] (20)

[0146] in, This is the cumulative trajectory length from this point to the starting point of the sub-segment. The total trajectory length of the sub-segment is calculated through numerical integration:

[0147] (twenty one)

[0148] Then decouple the corresponding mapping relationship and construct and An independent mapping table is used to achieve variable decoupling.

[0149] 4) Construct the Kriging proxy model

[0150] First, define the input and output of the proxy model, where the input parameters are uncertain parameters. ,in Represents an n-dimensional real number space. The output parameters are each... corresponding speed and amplitude .

[0151] Assume the output follows a Gaussian process. :

[0152] (twenty two)

[0153] The mean function Typically, a constant or linear term is taken, and the covariance function is used. Select Gaussian kernel:

[0154] (twenty three)

[0155] Then, the hyperparameters are optimized using maximum likelihood estimation. ,in To output the variance, Given the nth dimension as the length scale, the optimized hyperparameters for:

[0156] (twenty four)

[0157] in The covariance matrix is ​​derived from the covariance function. calculate. For training data, The mean function value (e.g.) or ).

[0158] For each trajectory length The values ​​correspond to the training of two Kriging models:

[0159] (25)

[0160] (26)

[0161] 5) Use a surrogate model to predict the uncertainty response boundary.

[0162] Generate using Monte Carlo simulation Latin hypercube sampling The trained Kriging model is used to predict the value of each sample. and Mapping. Then, boundary statistics are performed on the rotational speed and amplitude for each fixed trajectory length. Values, calculate the extreme values ​​of the response for all Monte Carlo samples:

[0163] Speed ​​boundary:

[0164] (27)

[0165] (28)

[0166] Amplitude boundaries:

[0167] (29)

[0168] 0) Perform an inverse coordinate transformation on the steady-state response boundary to obtain the steady-state response boundary in the original coordinate system.

[0169] Will and The coordinate system is restored to its original position according to the length of each segment trajectory (the original coordinate system refers to the steady-state response solution obtained after solving the dynamic equation using the complex exponential harmonic balance method after the first sampling, specifically the speed-amplitude plane coordinate system used to initially construct the response curve). For each segment... ,according to Calculate the corresponding rotational speed and amplitude :

[0170] (30)

[0171] (31)

[0172] This invention proposes a rapid analysis method for the dynamic characteristics of a three-dimensional asymmetric rotor system for excitation rotors, and puts forward an efficient method for quantifying the uncertainty of steady-state response. The method includes verification on an asymmetric shaft response test experimental platform, Kriging Latin hypercube sampling, dynamic response decoupling coordinate transformation, and establishing a surrogate model search boundary. This method solves the problem of false resonance peaks that are easy to occur when using interval methods for uncertainty analysis of asymmetric rotor steady-state response, and at the same time reduces the number of samples required in the uncertainty quantification process.

[0173] By comparing the scanning method, tensor product method, and Latin hypercube sampling method, the effectiveness of the dynamic response decoupling coordinate method and the efficiency of the Latin hypercube sampling method proposed in this invention are demonstrated. Specifically, in one-dimensional uncertainty problems, the tensor product method saves 98.75% of the computation time compared to the scanning method. In three-dimensional and eight-dimensional uncertainty problems, the Latin hypercube sampling method further saves 44% and 99.75% of the computation time compared to the tensor product method, 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. Furthermore, it eliminates the spurious resonance peak phenomenon in the aforementioned uncertainty problems, broadening the applicability of the interval method.

[0174] Based on the above inventive concept, this invention proposes an embodiment to specifically illustrate and verify a method for analyzing the uncertainty steady-state response of a three-dimensional asymmetric rotor:

[0175] This example aims to verify the superiority of the motion response decoupled coordinates proposed in this invention, as well as the efficiency and accuracy of Kriging-based Latin hypercube sampling. The scanning method's sampling strategy involves sampling the interval parameters of each dimension at equal intervals, then combining the sample points across all dimensions into a tensor product space. By performing deterministic analysis on each sample point in this space, the steady-state response of all sample points is obtained, and finally, statistical analysis is performed to obtain the response boundary. The scanning method is a very robust approach that consistently yields accurate results; therefore, it is used in this example to validate the new method.

[0176] This invention considers the elastic modulus and density of the excitation rotor, the principal stiffness coefficient, principal damping coefficient, and anisotropy parameters of the bearings, as well as the relative angles between bearings related to installation errors, as uncertain parameters. These uncertain parameters not only affect the steady-state response of the system but also its stability. This section studies seven combinations of uncertain parameters, and the fluctuation coefficients under all conditions are shown in Table 1.

[0177] Table 1 Uncertainty parameters under different operating conditions

[0178]

[0179] First, taking the above example as an example, we study the application of motion response decoupling coordinates in the analysis of the uncertainty steady-state response of the excitation rotor. In this case, the fluctuation coefficient of the elastic modulus is 4%. Figure 6 and Figure 7 The interval response results of isotropic systems obtained using the scanning method and the tensor product method are presented respectively. Figure 6 As shown, the interval response results with 400 and 800 samples agree well, indicating that the interval response results with 400 samples have converged. Figure 7 As shown in (a), using the tensor product method based on rectangular coordinates, when the number of sampling points is 5, spurious resonance peaks appear at the 3rd resonance peak, the 2nd anti-resonance peak, and the 5th resonance peak in the response curve. This means that these resonance peaks are not caused by parameter uncertainties, but rather by the polynomial surrogate model's incorrect prediction of the steady-state response. Resonance bands only appear at the last three peaks because the 1st and 2nd resonance peaks are not very sensitive to the elastic modulus and do not show a shift in resonance frequency, while the 3rd, 4th, and 5th resonance peaks are more sensitive to the elastic modulus and exhibit wider resonance bands. However, because the amplification factor of the 4th resonance peak is small, no obvious spurious resonance peak appears. Similarly, the 2nd anti-resonance peak is highly sensitive to the elastic modulus and therefore also shows an obvious spurious resonance peak. From the above analysis, it can be seen that spurious resonance peaks tend to appear at resonance peaks with large amplification factors. Figure 7 (b) to Figure 7 As shown in (d), increasing the number of sampling points to 10, 15, and 20 respectively reveals that the spurious resonance peak phenomenon improves with the increase in the number of sampling points. Specifically, when the number of sampling points increases to 10, the spurious resonance peaks at the 3rd and 5th resonance peaks essentially disappear. However, even when the number of sampling points increases to 20, the spurious resonance peak at the 2nd anti-resonance peak can only be improved, not completely eliminated. This is mainly because the response amplitude at this resonance peak changes drastically, making it difficult to describe using a polynomial.

[0180] Figure 8 This paper presents the interval response and boundary error-speed curves of an isotropic system obtained using the tensor product method based on motion response decoupling coordinates. It can be observed that the application of motion response decoupling coordinates eliminates spurious resonance peaks at each resonance and anti-resonance peak. The results obtained using the tensor product method based on motion response decoupling coordinates are compared with those obtained using the scanning method (…). Figure 6The results are very consistent, with prediction errors mostly not exceeding 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. The tensor product method based on motion response decoupled coordinates can save 98.75% of the computation time and improve computational efficiency.

[0181] Based on the same inventive concept, this invention also proposes a three-dimensional asymmetric excitation rotor uncertainty steady-state response analysis system, comprising:

[0182] The dynamic model construction module is used to establish a dynamic model of the excitation rotor system containing interval uncertainty based on the three-dimensional finite element dynamic model of the excitation rotor by introducing interval uncertainty parameters.

[0183] The solution module is used to map uncertain parameters in the dynamic model of the excitation rotor system to a standard hypercube space using the Latin hypercube sampling method, generating a Latin hypercube sampling sample space. Response analysis is performed on each sample point in the sample space to establish a deterministic steady-state response curve for each sample point. Extrema detection is performed on the deterministic steady-state response curve, and the curve is divided into multiple sub-segments based on the detected extrema. Rotational speed and response amplitude are decoupled in dynamic response coordinates, and the decoupling coordinates for each sub-segment are determined. 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 rotational speed and response amplitude. Finally, a Kriging model is established based on the uncertain parameters and the decoupled independent mapping relationship.

[0184] The generation module is used to generate a new Latin hypercube sampling sample space using Monte Carlo simulation. By establishing the Kriging model, it predicts the mapping relationship between the dynamic response coordinates of each sample in the sample space and the rotational speed and response amplitude, and generates a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system.

[0185] The analysis module is used to analyze the uncertain 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.

[0186] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor, characterized in that, Includes the following steps: Based on the three-dimensional asymmetric excitation rotor, a three-dimensional finite element dynamic model is established by introducing interval uncertainty parameters to create a dynamic model of the excitation rotor system containing interval uncertainty. The Latin hypercube sampling method is used to map the uncertain parameters in the dynamic model of the excitation rotor system to a standard hypercube space, generating a Latin hypercube sampling sample space. Response analysis is performed on each sample point in the sample space to establish a deterministic steady-state response curve for each sample point. Extreme value detection is performed on the deterministic steady-state response curve, and the curve is divided into multiple sub-segments based on the detected extreme points. The rotational speed and response amplitude are decoupled in the dynamic response coordinate system, and the decoupling coordinates for each sub-segment are determined. Based on the decoupling coordinates of each sub-segment, the deterministic steady-state response curve is transformed into an independent mapping relationship between the dynamic response coordinates and the rotational speed and response amplitude. Based on the uncertain parameters and the decoupled independent mapping relationship, a Kriging model is established. A new Latin hypercube sampling sample space is generated using Monte Carlo simulation. The mapping relationship between the dynamic response coordinates of each sample in the sample space and the rotational speed and response amplitude is predicted by the established Kriging model, thereby 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 characterizing the uncertainty of the excitation rotor system, the uncertain steady-state response of the three-dimensional asymmetric excitation rotor is analyzed.

2. The method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1, characterized in that, The three-dimensional finite element dynamic model based on a three-dimensional asymmetric excitation rotor establishes a dynamic model of the excitation rotor system containing interval uncertainties by introducing interval uncertainty parameters. Specifically, the model includes the following steps: Establish a three-dimensional finite element dynamic model of a three-dimensional asymmetric excitation rotor: ； Define the presence of excitation rotor system n u If there are several interval uncertainty parameters, then the interval vector is represented as: ； Represent the interval vector as a sum of the midpoint and the interval radius: ； in: ； In the formula: It is an interval vector; , Let these be the lower and upper bound vectors of the interval; It is the midpoint of the interval vector; Let be the radius of the interval vector; Based on the three-dimensional finite element dynamic model of the excitation rotor, the dynamic equations of the excitation rotor system containing interval uncertainties are obtained as follows: ； in, Here is the mass matrix of the excitation rotor; The Coriolis damping matrix of the excitation rotor; This represents the static stiffness matrix of the excitation rotor; For the rotating softening matrix of the excitation rotor; For the displacement response of the bearing system of a two-pole generator; The velocity vector of the two-pole generator bearing system; The acceleration vector of the two-pole generator bearing system; It is the gravity vector; This is the cosine modulation term for bearing stiffness; This is the sinusoidal modulation term for bearing stiffness; , Here is the Coriolis damping matrix of the rotor. Let be the static stiffness matrix of the rotor. Here is the dynamic stiffness matrix of the rotor. This represents the static stiffness component of the bearing. The fundamental angular frequency of the excitation.

3. The method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1, characterized in that, The process of detecting extreme values ​​in the deterministic steady-state response curve and dividing the curve into multiple sub-segments based on the detected extreme points includes the following steps: Extreme points are marked on the deterministic steady-state response curve, and the decoupling coordinates of the motion response at the starting point are initialized to 0; When traversing along the trajectory of the deterministic steady-state response curve, the value of the motion response decoupling coordinate is increased by 1 unit for each extreme point detected, until the endpoint of the deterministic steady-state response curve is reached, and then increased by 1 unit again. The deterministic steady-state response curve is divided into several segments based on the identified starting point, extreme point, and ending point.

4. The method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1, characterized in that, The method employs Latin hypercube sampling to map uncertain parameters in the dynamic model of the excitation rotor system to a standard hypercube space, generating a Latin hypercube sampling sample space; specifically, it includes the following steps: Let the uncertainty parameter in the dynamic equation of the excitation rotor system containing interval uncertainty be: Each parameter follows a uniform distribution Mapping parameters to standard hypercube space The normalized mapping is: ； in, a i , b i This refers to each uncertainty parameter The upper and lower limits; Divide the dimensional subspace, dividing each dimension into... Each sub-interval has equal width. Within each sub-interval, a sample point is randomly selected and the interval order of each dimension is randomly arranged. The sample points of each dimension are randomly arranged and combined, and the number of samples is: ； in, The order of the surrogate model is truncated. As an dimension of uncertainty; Transfer sample points from Mapping back to the original parameter range, we get: ； The final Latin hypercube sampling point set is as follows .

5. The method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor according to claim 4, characterized in that, The process of performing response analysis on each sample point in the sample space and establishing a deterministic steady-state response curve for each sample point includes the following steps: For each sample point The steady-state response of the excitation rotor system dynamics equations containing interval uncertainties is solved using the complex exponential harmonic balance method: ； Assuming the steady-state response is a periodic function, it can be represented using a complex exponential series as follows: ； in, The order of the harmonics. The harmonic cutoff order is... The complex amplitude of each harmonic component in response, For the fundamental angular frequency of the excitation, To determine the complex amplitude of each harmonic component in the excitation; Let be the displacement response vector of the generator bearing system with respect to time; Regarding time t The gravity vector; Substituting the expansion of the steady-state response into the dynamic equations and matching the coefficients of each harmonic term, we obtain a system of linear equations: ； in, The mass matrix, which is the harmonic order. inertial term, For the solution of motion response; The domain coefficients can be solved by matrix inversion or iterative methods. Reconstructing the time-domain response yields the deterministic steady-state response curve for each sample point: ； in The number of discrete points for rotational speed. For each sample point of the response curve, the rotational speed value. The response amplitude for each sample point, the vibration amplitude is .

6. The method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor according to claim 3, characterized in that, The process of establishing a Kriging model based on uncertainty parameters and decoupled independent mapping relationships includes the following steps: Define the input parameters as uncertain parameters. ,in express n The output parameter is the length of each sub-segment trajectory in a 3D real space. corresponding speed and amplitude ; The output parameter is set to follow a Gaussian process. : ； in, The covariance function for controlling rotational speed. The covariance function and mean function are used to control the response amplitude. Take a constant or linear term, covariance function Using a Gaussian kernel, we get: ； Optimize hyperparameters using maximum likelihood estimation. ,in To output the variance, For the first n Dimension length scale, optimized hyperparameters for: ； in The covariance matrix is ​​derived from the covariance function. calculate; For training data, , , The value is the mean function value; For each trajectory length The Kriging models were trained accordingly: ； 。 7. The method for analyzing the uncertain steady-state response of a three-dimensional asymmetric excitation rotor according to claim 1, characterized in that, The method utilizes 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 rotational speed and response amplitude, thereby generating a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system. Specifically, the following steps are included: A new Latin hypercube sampling set was generated using Monte Carlo simulation. By establishing the Kriging model, the mapping relationship between the trajectory length of the deterministic steady-state response curve of each sample and the rotational speed and response amplitude is predicted, and the rotational speed and amplitude boundaries of each sample are determined. Based on the segment trajectory length, the boundaries of rotational speed and amplitude are mapped back to the original deterministic steady-state response curve, generating a steady-state response envelope curve 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 build a three-dimensional finite element dynamic model based on a three-dimensional asymmetric excitation rotor. By introducing interval uncertainty parameters, a dynamic model of the excitation rotor system containing interval uncertainty is established. The solution module is used to map uncertain parameters in the dynamic model of the excitation rotor system to a standard hypercube space using the Latin hypercube sampling method, generating a Latin hypercube sampling sample space. Response analysis is performed on each sample point in the sample space to establish a deterministic steady-state response curve for each sample point. Extrema detection is performed on the deterministic steady-state response curve, and the curve is divided into multiple sub-segments based on the detected extrema. Rotational speed and response amplitude are decoupled in dynamic response coordinates, and the decoupling coordinates for each sub-segment are determined. 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 rotational speed and response amplitude. Based on the uncertain parameters and the decoupled independent mapping relationship, a Kriging model is established. The generation module is used to generate a new Latin hypercube sampling sample space using Monte Carlo simulation. By establishing the Kriging model, it predicts the mapping relationship between the dynamic response coordinates of each sample in the sample space and the rotational speed and response amplitude, and generates a steady-state response envelope curve that can characterize the uncertainty of the excitation rotor system. The analysis module is used to analyze the uncertain 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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