HB-AFT-based shell-rotor system model rapid solving method

Abstract

The invention discloses an HB-AFT-based shell-rotor system model rapid solving method, which comprises the following steps of: establishing a kinetic model of a complex shell-rotating shaft-motor coupling system, firstly performing primary model order reduction based on an intrinsic orthogonal decomposition (POD) method, and then performing model rapid solving based on the HB-AFT method. According to the method, the rotor system has higher calculation efficiency, the complex nonlinear vibration problem can be rapidly solved, and the dynamic behavior mechanism of the core cabin of the complex underwater vehicle can be clarified. Compared with a traditional numerical method, the method has the advantages that an unstable solution can be obtained through the semi-analysis and semi-numerical method, so that researchers are helped to better analyze the vibration phenomenon of the underwater vehicle core cabin system, and then optimization design is conducted on the underwater vehicle core cabin system.

Description

Technical Field

[0001] The present invention belongs to the technical field of dynamics, and in particular relates to a method for quickly solving a shell-rotor system model based on HB-AFT. Background Art

[0002] The core compartment of an underwater vehicle is an internal high-speed rotating power system, a complex coupled system consisting of a shell, a rotating shaft, a motor, and ancillary components. The shell-rotor system has numerous ancillary components and connection structures, and is characterized by a high number of degrees of freedom, strong coupling, and strong nonlinearity. In actual engineering, the common method for solving the vibration response of large and complex systems is to simulate using finite element, finite difference, finite volume, and other methods. As research deepens, the requirements for model solution accuracy have increased significantly. The degrees of freedom of refined models established using finite element and other methods are tens of thousands or even hundreds of millions, and it is difficult to converge the numerical iterative solution of strongly nonlinear systems, resulting in a single calculation time of tens of hours or even longer based on existing computing power. Therefore, there is an urgent need to develop a fast solution method suitable for strongly nonlinear systems. Summary of the Invention

[0003] In order to overcome the shortcomings of the prior art, the present invention provides a method for quickly solving the shell-rotor system model based on HB-AFT, establishes a dynamic model of a complex shell-shaft-motor coupling system, first performs a first-order model reduction based on the proper orthogonal decomposition (POD) method, and then quickly solves the model based on the HB-AFT method. The present invention enables the rotor system to have higher computational efficiency, can quickly solve complex nonlinear vibration problems, and elucidate the dynamic behavior mechanism of the core compartment of a complex underwater vehicle. Compared with traditional numerical methods, the present invention uses a semi-analytical and semi-numerical method to obtain unstable solutions, thereby helping researchers better analyze the vibration phenomenon of the core compartment system of the underwater vehicle, and then optimize the design of the core compartment system of the underwater vehicle.

[0004] The technical solutions adopted by the present invention to solve the technical problems are as follows:

[0005] Step 1: Establish a dynamic model of the housing-rotor system;

[0006] Step 2: Model reduction based on POD method;

[0007] Step 3: Fast solution based on HB-AFT method.

[0008] Preferably, the step 1 is specifically:

[0009] The damping of the housing-rotor system is Rayleigh damping. Considering the Hertzian contact force of the bearings, the expression of the dynamic differential equation of the housing-rotor system is as follows:

[0010]

[0011] Where M, C, G, and K are the total mass, damping, gyro, and stiffness matrices of the system, respectively; C = αM + βK, where α and β are constants, ω is the motor speed, u is the displacement vector, F is the generalized external force vector, and F = F g +F e +F b +F r , F g is gravity, F e is the eccentric force of the motor and the weight, F b is the bearing Hertzian contact force, F r It is external motivation;

[0012] The Hertz contact force expression is as follows:

[0013]

[0014] Among them, θ j is the instantaneous angle of the jth rolling element, δ j is the contact deformation between the jth rolling element and the bearing ring, d is the Hertz contact nonlinear index, H(δ ij ) is the Heaviside function of the bearing, C b 、N b , G are Hertz contact stiffness coefficient, number of rollers and radial clearance of bearing respectively; x j Indicates the x-axis coordinate of the j-th rolling element, y j Indicates the y-axis coordinate of the j-th rolling element, F bxj represents the x-axis component of the force between the jth rolling element and the bearing ring, F byj It represents the y-axis component of the force between the j-th rolling element and the bearing ring.

[0015] Preferably, the step 2 is specifically as follows:

[0016] Step 2-1: Calculate the response signal of formula (1) by numerical method in is the free coordinate vector of the dynamic system, t is the time variable, is the system response; assuming that the response of the n-degree-of-freedom dynamic system is written as:

[0017]

[0018] where a i (t) is the i-th order mode function, is the i-th order modal vector, and all modal vectors form a set of complete orthogonal bases;

[0019] Step 2-2: Find a set of optimal orthogonal bases as the dimensionality reduction orthogonal bases using formula (4);

[0020]

[0021] where <·> represents the averaging operator, (·,·) represents the inner product on the Hilbert space, and ‖·‖ 2 Indicates L 2 norm, represents the modal vector;

[0022] Step 2-3: Obtain the optimal orthogonal basis solution of formula (4) by the Lagrange multiplier method of formula (5);

[0023]

[0024] Where J is the Lagrange function and λ is the Lagrange multiplier;

[0025] Integrating formula (5) yields:

[0026]

[0027] Formula (6) is the second kind of Fredholm equation, and its kernel function is: The characteristic root λ of formula (6) and its corresponding characteristic vector At the same time Discrete time series The constructed autocorrelation matrix The eigenvalues ​​and eigenvectors of The number of discrete time series samples; by finding the autocorrelation matrix The eigenvalues ​​and eigenvectors of are used to obtain a set of optimal orthogonal bases found by formula (4);

[0028] Step 2-4: Project formula (3) onto the space spanned by the eigenvectors corresponding to the first m largest eigenvalues, and obtain:

[0029]

[0030] Coordinate transformation of formula (7):

[0031]

[0032] Step 2-5: Combining formula (8) and formula (1), we can obtain the dimensionality reduction system dynamics differential equation:

[0033]

[0034] make Then we get the simplified equation (9):

[0035]

[0036] The original system is reduced from n degrees of freedom to m degrees of freedom.

[0037] Preferably, the step 3 is specifically as follows:

[0038] Step 3-1: Solve formula (10) through formula (11):

[0039]

[0040] in, for The harmonic coefficient of , s is the number of retained harmonics, The nonlinear force The harmonic term coefficient, A, is calculated by formula (14);

[0041]

[0042] in, T is the excitation frequency period;

[0043]

[0044] Step 3-2: Solve formula (11) using the Newton-Raphson iterative method of formula (15);

[0045]

[0046] in, represents the approximate solution obtained in the i-th iteration step, Represents the Newton iteration formula The approximate solution of the next iteration point obtained after updating is The objective equation is expressed as The function value at ; J is the Jacobian matrix, in By formula (16), we can get:

[0047]

[0048] The terms are obtained by the chain derivation rule of formulas (17)-(22):

[0049]

[0050] Similarly:

[0051]

[0052] Set initial value Newton-Raphson iteration termination condition The result is obtained by Newton-Raphson iteration The iterative results Perform inverse Fourier transform to obtain an approximate analytical solution to the reduced-dimensional system

[0053] Preferably, the system comprises a housing, a motor, a flange and a weight.

[0054] Preferably, the Hertzian contact nonlinear index of the ball bearing is d=3 / 2, and the Hertzian contact nonlinear index of the roller bearing is d=10 / 9.

[0055] A computer program enables a computer to execute the above-mentioned method for quickly solving the model.

[0056] An electronic device comprises: a processor and a memory; the memory is used to store a computer program, and the processor is used to execute the computer program stored in the memory, so that the electronic device executes the above-mentioned model rapid solution method.

[0057] A computer-readable storage medium stores a computer program, which implements the above-mentioned model fast solution method when executed by a processor.

[0058] A chip includes: a processor for calling and running a computer program from a memory, so that a device equipped with the chip executes the above-mentioned model rapid solution method.

[0059] The beneficial effects of the present invention are as follows:

[0060] The present invention enables the rotor system to have higher computational efficiency during pattern expansion, and can quickly clarify the changing laws of the dynamic behavior of complex underwater vehicle rotor systems, thereby helping researchers to better analyze the vibration phenomena of the engine turbine rotor-bearing system and further optimize the design of the underwater vehicle shell-rotor system. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 It is a schematic diagram of the fast model solving method combining the POD method and the HB-AFT method of the present invention;

[0062] Figure 2 is a schematic diagram of the dynamic model of the housing-rotor system of the present invention;

[0063] Figure 3This is a comparison diagram of the time history curves and spectra of the original system and the simplified system of the present invention. Figure (a) is the time history curve in the horizontal x direction, Figure (b) is the spectrum in the horizontal x direction, Figure (c) is the time history curve in the vertical z direction, and Figure (d) is the spectrum in the vertical z direction.

[0064] Figure 4 This is a comparison diagram of the time history curves and spectra solved by the simplified system using the fourth-order classical Runge-Kutta method and the HB-AFT method. Figure (a) is the time history curve in the horizontal x direction using the fourth-order classical Runge-Kutta method, and Figure (b) is the spectrum in the horizontal x direction using the fourth-order classical Runge-Kutta method. Figure (c) is the time history curve in the vertical z direction using the HB-AFT method, and Figure (d) is the spectrum in the vertical z direction using the HB-AFT method.

[0065] Figure 5 It is a comparison diagram of the axis trajectory. Figure (a) is a comparison diagram of the axis trajectory of the original system and the simplified system. Figure (b) is a comparison diagram of the axis trajectory of the fourth-order classical Runge-Kutta method and the HB-AFT method.

[0066] Description of the drawings: 1-components, 2-bearings and bearing seats, 3-motor, 4-housing, 5-weight, 6-flange. DETAILED DESCRIPTION

[0067] The present invention will be further described below with reference to the accompanying drawings and examples.

[0068] 1. Establish a dynamic model of the housing-rotor system;

[0069] The damping of the housing-rotor system is Rayleigh damping. Considering the Hertzian contact force of the bearings, the expression of the dynamic differential equation of the housing-rotor system is as follows:

[0070]

[0071] Where M, C, G, and K are the total mass of the system (the system includes the housing, motor, flange, weight and other accessories), damping, gyroscope, and stiffness matrices, respectively. C = αM + βK, where α and β are constants, ω is the speed, u is the displacement vector, and F is the generalized external force vector. F = F g +F e +F b +F r , F g is gravity, F e is the eccentric force of the motor and the weight, F b is the bearing Hertzian contact force, F r It is external motivation.

[0072] The Hertz contact force expression is as follows:

[0073]

[0074] Among them, θ j is the instantaneous angle of the jth rolling element, δ j is the contact deformation between the jth rolling element and the bearing ring, d is the Hertz contact nonlinear index (d = 3 / 2 for ball bearings, d = 10 / 9 for roller bearings), H(δ j ) is the Heaviside function of the bearing, C b , N b , G are the Hertzian contact stiffness coefficient, the number of rollers and the radial clearance of the bearing respectively.

[0075] 2. Model reduction based on POD method;

[0076] The POD method is used to reduce the order of the shell-rotor system model. The detailed steps are as follows:

[0077] The response signal of formula (1) is solved numerically in is the free coordinate vector of the dynamic system, t is the time variable, Assume that the response of the n-DOF dynamic system can be written as: a i (t) is the i-th order mode function, is the i-th order modal vector, and all modal vectors form a set of complete orthogonal bases.

[0078] Under given initial conditions, for each degree of freedom u1(t), u2(t),…,u n (t), all have transient displacement responses, n is the total degree of freedom of the system. The transient displacement response of the i-th degree of freedom is expressed as The N time series are sampled at equal intervals. This gives an N×n signal matrix X, which is shown as:

[0079]

[0080] Construct the autocorrelation matrix R=U of U T U, and find its eigenvalue and eigenvector. The eigenvalue and the corresponding eigenvector can be expressed as a1>a2>…a n ≥0 and Where N represents The number of discrete time series samples. Projecting the original system onto the space spanned by the eigenvectors corresponding to the first m largest eigenvalues ​​yields:

[0081]

[0082] Coordinate transformation of formula (5):

[0083] Combining formula (5) and formula (1), we get the dimensionality reduction system dynamics differential equation:

[0084]

[0085] make Then we get the simplified equation (6):

[0086]

[0087] The original system is reduced from n degrees of freedom to m degrees of freedom.

[0088] 3. Fast solution based on HB-AFT method;

[0089] Apply the HB-AFT method to solve formula (7). The detailed steps are as follows:

[0090] Solve formula (7) through formula (8):

[0091]

[0092] in, for The harmonic coefficient of , s is the number of retained harmonics, ω is the excitation frequency (Formula (9)), The nonlinear force The harmonic coefficient A is calculated by formula (11).

[0093]

[0094]

[0095] in, T is the excitation frequency period.

[0096]

[0097] Formula (8) can be solved by the Newton-Raphson iterative method (Formula (12)):

[0098]

[0099] J is the Jacobian matrix of the formula, in Calculated by formula (13);

[0100]

[0101] The various terms are obtained by the chain rule (Formulas (14)-(19)):

[0102]

[0103] Similarly, we can get:

[0104]

[0105] Set initial value Newton-Raphson iteration termination condition The result is obtained by Newton-Raphson iteration The iterative results Perform inverse Fourier transform to obtain an approximate analytical solution to the reduced-dimensional system

[0106] 4. Data analysis and verification method reliability;

[0107] Figures 3 to 5 The corresponding results are given, which verify the effectiveness of the POD method and HB-AFT method in solving the model response. Figure 3 The time history curve and spectrum comparison diagram of the original system and the simplified system are shown in Figure 2. Figure 3 (a) (b) are the time history curve and spectrum in the horizontal x direction. Figure 3 (c) and (d) show the time history curves and frequency spectra in the vertical z-direction. As can be seen, the time history curves of the original and simplified systems have minimal error in the vertical and horizontal directions, and the time history curves of the original and simplified systems almost completely overlap, indicating that the simplified system retains the vast majority of information. When describing the dynamic behavior of the original system, the spectrum of the simplified system fully preserves the system's frequency components, with amplitudes nearly identical. The amplitude of the main frequency of the simplified system is highly consistent with that of the original system.

[0108] Figure 4 In order to simplify the system, the time history curve and spectrum comparison diagram solved by the fourth-order classical Runge-Kutta method and HB-AFT method are shown. Figure 4 (a) (b) are the time history curve and spectrum in the horizontal x direction. Figure 4 (c) and (d) show the time history curves and spectra in the vertical z-direction. As can be seen from the figures, the time history curves for the Runge-Kutta method and the HB-AFT method in the vertical and horizontal directions are very close overall, with consistent periods and minimal error, as evidenced by the near-complete overlap of the time history curves for the two methods. The HB-AFT method also demonstrates high accuracy in describing the system's frequency information, with the spectra of the two methods nearly overlapping.

[0109] Figure 5This is a comparison chart of the axis trajectory. Figure 5 (a) is a comparison diagram of the axis trajectory of the original system and the simplified system. Figure 5 (b) is a comparison of the axis trajectory of the fourth-order classical Runge-Kutta method and the HB-AFT method. As can be seen from the figure, the axis trajectory of the system is a closed ellipse; Figure 5 (a) The axis trajectory of the original system and the simplified system are very close overall, and the curve amplitudes almost coincide with each other, with a small error. This is manifested in that the axis trajectory curve of the simplified system is smaller than that of the original system at the minor axis of the elliptic curve. This is because the use of the POD method to reduce the model loses some system information; Figure 5 In (b), the axis trajectory curves of the two methods almost completely overlap, with extremely small errors, indicating that the HB-AFT method has high accuracy.

[0110] In summary, the POD method combined with the HB-AFT method offers high accuracy in solving model responses. The POD method retains the vast majority of system information. Furthermore, the HB-AFT semi-analytical, semi-numerical method replaces the fourth-order classical Runge-Kutta method, significantly reducing computational costs while maintaining high accuracy. Therefore, the HB-AFT fast solution algorithm of the present invention can effectively and rapidly solve large, complex systems with high accuracy.

Claims

1. A fast solution method for the shell-rotor system model based on HB-AFT, characterized by: The steps include: Step 1: Establish a dynamic model of the housing-rotor system; Step 2: Model reduction based on POD method; Step 3: Fast solution based on HB-AFT method.

2. The method for quickly solving the housing-rotor system model based on HB-AFT according to claim 1, characterized in that: The step 1 is specifically as follows: The damping of the housing-rotor system is Rayleigh damping. Considering the Hertzian contact force of the bearings, the expression of the dynamic differential equation of the housing-rotor system is as follows: Where M, C, G, and K are the total mass, damping, gyro, and stiffness matrices of the system, respectively; C = αM + βK, where α and β are constants, ω is the motor speed, u is the displacement vector, F is the generalized external force vector, and F = F g +F e +F b +F r , F g is gravity, F e is the eccentric force of the motor and the weight, F b is the bearing Hertzian contact force, F r It is external motivation; The Hertz contact force expression is as follows: Among them, θ j is the instantaneous angle of the jth rolling element, δ j is the contact deformation between the jth rolling element and the bearing ring, d is the Hertz contact nonlinear index, H(δ ij ) is the Heaviside function of the bearing, C b 、N b , G are Hertz contact stiffness coefficient, number of rollers and radial clearance of bearing respectively; x j Indicates the x-axis coordinate of the j-th rolling element, y j Indicates the y-axis coordinate of the j-th rolling element, F bxj represents the x-axis component of the force between the jth rolling element and the bearing ring, F byj It represents the y-axis component of the force between the j-th rolling element and the bearing ring.

3. The method for quickly solving the housing-rotor system model based on HB-AFT according to claim 2, characterized in that: The step 2 is specifically as follows: Step 2-1: Calculate the response signal of formula (1) by numerical method in is the free coordinate vector of the dynamic system, t is the time variable, is the system response; assuming that the response of the n-degree-of-freedom dynamic system is written as: where a i (t) is the i-th order mode function, is the i-th order modal vector, and all modal vectors form a set of complete orthogonal bases; Step 2-2: Find a set of optimal orthogonal bases as the dimensionality reduction orthogonal bases using formula (4); where <·> represents the averaging operator, (·,·) represents the inner product on the Hilbert space, and ‖·‖ 2 Indicates L 2 norm, represents the modal vector; Step 2-3: Obtain the optimal orthogonal basis solution of formula (4) by the Lagrange multiplier method of formula (5); Where J is the Lagrange function and λ is the Lagrange multiplier; Integrating formula (5) yields: Formula (6) is the second kind of Fredholm equation, and its kernel function is: The characteristic root λ of formula (6) and its corresponding characteristic vector At the same time Discrete time series Autocorrelation matrix constructed with j=1,2,…,N The eigenvalues ​​and eigenvectors of The number of discrete time series samples; by finding the autocorrelation matrix The eigenvalues ​​and eigenvectors of are used to obtain a set of optimal orthogonal bases found by formula (4); Step 2-4: Project formula (3) onto the space spanned by the eigenvectors corresponding to the first m largest eigenvalues, and obtain: Coordinate transformation of formula (7): Step 2-5: Combining formula (8) and formula (1), we can obtain the reduced-dimensional system dynamics differential equation: make Then we get the simplified equation (9): The original system is reduced from n degrees of freedom to m degrees of freedom.

4. The method for quickly solving the housing-rotor system model based on HB-AFT according to claim 3, characterized in that: The step 3 is specifically as follows: Step 3-1: Solve formula (10) through formula (11): in, for The harmonic coefficient of , s is the number of retained harmonics, The nonlinear force The harmonic term coefficient, A, is calculated by formula (14); in, k=1,2,…,s, T is the excitation frequency period; Step 3-2: Solve formula (11) by the Newton-Raphson iterative method of formula (15); in, represents the approximate solution obtained in the i-th iteration step, Represents the Newton iteration formula The approximate solution of the next iteration point obtained after updating is The objective equation is expressed as The function value at ; J is the Jacobian matrix, in By formula (16), we can get: The terms are obtained by the chain derivation rule of formulas (17)-(22): Similarly: Set initial value Newton-Raphson iteration termination condition The result is obtained by Newton-Raphson iteration The iterative results Perform inverse Fourier transform to obtain an approximate analytical solution to the reduced-dimensional system 5. The method for quickly solving the housing-rotor system model based on HB-AFT according to claim 4, characterized in that: The system includes a shell, a motor, a flange and a weight.

6. The method for quickly solving the housing-rotor system model based on HB-AFT according to claim 5, characterized in that: The Hertzian contact nonlinear index of the ball bearing is d=3 / 2, and the Hertzian contact nonlinear index of the roller bearing is d=10 / 9.

7. A computer program, characterized in that The computer program enables a computer to execute the method according to any one of claims 1 / 6.

8. An electronic device, characterized in that: include: processor and memory; The memory is used to store a computer program, and the processor is used to execute the computer program stored in the memory, so that the electronic device performs the method according to any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the method according to any one of claims 1 to 6 is implemented.

10. A chip, characterized in that: include: A processor, configured to call and run a computer program from a memory, so that a device equipped with the chip executes the method according to any one of claims 1 to 6.

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