A method for analyzing stability of an inertial asymmetric rotor system

Abstract

This application discloses a stability analysis method for an inertial asymmetric rotor system. The method includes: 1) modeling the dynamic equations of the inertial asymmetric rotor system; 2) identifying key influencing factors of the difference between rotational speed and inertia as parametric excitation terms, and considering the difference in inertia as a disturbance term; 3) decoupling the two ends of the dynamic equations using real modal analysis; 4) performing power series expansion of the degrees of freedom and stability boundaries using the perturbation method and substituting them into the differential equation system; 5) solving the differential equation system sequentially to obtain the periodic solution of the inertial asymmetric rotor system response, the system's instability interval, and the analytical solution of the vibration response. This application considers that the system's rotational inertia is time-varying, i.e., it considers the parametric excitation problem of the inertial asymmetric rotor. By using real modal analysis and the perturbation method, the analytical solution of the vibration response is obtained, and the instability interval of the inertial asymmetric rotor system is obtained, which improves the computational efficiency and accuracy and provides theoretical calculation guidance for rotor containment tests involving parametric excitation instability.

Description

Technical Field

[0001] This application belongs to the field of rotating machinery vibration analysis technology, specifically relating to a stability analysis method for an inertial asymmetric rotor system. Background Technology

[0002] To save costs and time during containment testing of aero-engine rotors, as few blades as possible can be installed on the shaft, resulting in a special rotor with an asymmetric inertial layout in the circumferential direction. In destructive testing, the dynamic response and stability of a rotor with a small number of blades undergo significant changes after dynamic balancing. Early designs of asymmetric inertial rotors revealed that the system could become unstable when the rotational speed is far below the target speed. Therefore, in-depth research into the dynamic response and stability of asymmetric inertial rotor systems is crucial to ensuring the successful conduct of related tests.

[0003] In the field of rotating machinery, inertial asymmetric rotor systems exhibit different inertia and stiffness on two principal inertial planes. When considering displacement and angular coupling vibrations caused by eccentricity, parametric excitation may occur. Parametric excitation induced by asymmetric inertial distribution can lead to instability and severe vibration under certain operating conditions. Therefore, research on the design and use of inertial asymmetric rotor systems involving parametric excitation instability is crucial in blade destructive testing.

[0004] In summary, there is an urgent need for a stability analysis method for inertial asymmetric rotor systems. Summary of the Invention

[0005] The purpose of this application is to provide a stability analysis method for inertial asymmetric rotor systems. This method can effectively solve the technical problem of rotor instability caused by parametric excitation induced by asymmetric inertial distribution under certain operating conditions.

[0006] To achieve the above objectives, this application provides a stability analysis method for an inertial asymmetric rotor system, mainly including:

[0007] Step S1: Establish the dynamic equation model of the inertial asymmetric rotor system;

[0008] Step S2: Decompose the dynamic equation model of the inertial asymmetric rotor system, place the key influencing factor terms containing the speed and inertia difference as parameter excitation terms on the right side of the dynamic equation, consider the inertia difference as a disturbance term, place the terms without parameter excitation terms on the left side of the dynamic equation, and write the dynamic equation in matrix form.

[0009] Step S3: Using the real modal analysis method, the left and right sides of the matrix-form dynamic equations are decoupled respectively to obtain the decoupled second-order differential equation system.

[0010] Step S4: Use the perturbation method to expand the degrees of freedom and stability boundaries in the decoupled differential equation system into power series and substitute them into the differential equation system. List the equation system according to the same power of the perturbation terms.

[0011] Step S5: Solve the equations sequentially to obtain the periodic solution, instability interval, and analytical solution of the vibration response of the inertial asymmetric rotor system.

[0012] Preferably, step S1 further includes:

[0013] Based on the inertial asymmetric rotor system model, the dynamic equations of the inertial asymmetric rotor system are modeled in a fixed coordinate system.

[0014] In the case of neglecting damping, the dynamic equation of the inertial asymmetric rotor system is:

[0015] ;

[0016] In the formula, For the quality matrix, It is a gyroscope matrix. It is the stiffness matrix. It is a generalized displacement vector.

[0017] Preferably, step S2 further includes:

[0018] By treating the key influencing factors of rotational speed and inertia difference as parametric excitation terms on the right-hand side of the dynamic equation, and considering the inertia difference as a disturbance term, while placing terms without parametric excitation terms on the left-hand side, the dynamic equation can be written in matrix form as follows:

[0019] ;

[0020] In the formula, The mass matrix contains no parameterized excitation terms. It is a stiffness matrix that does not contain parametric excitation terms. It is a mass matrix containing parameter excitation terms. It is a gyroscope matrix containing parameter excitation terms.

[0021] Preferably, step S3 further includes:

[0022] Let the vector formed by the characteristic roots of the left-hand side of the dynamic equation be... , The eigenvector corresponding to each eigenvalue is ,in Arrange the eigenvectors into a 4x4 modality matrix. By decoupling the mass matrix M (without parameter excitation terms) and the stiffness matrix K (without parameter excitation terms) on the left-hand side of the dynamic equation using this modal matrix, the expression is obtained as follows:

[0023] ;

[0024] In the formula, The modal principal mass matrix is... Given the modal principal stiffness matrix, the fundamental frequencies of the model are obtained by solving for the model's fundamental stiffness matrix.

[0025] The same transformation is applied to the mass matrix M' containing the parametric excitation term and the gyroscope matrix G' containing the parametric excitation term on the right-hand side of the dynamic equation, resulting in matrices P and Q, respectively:

[0026] ;

[0027] Due to the independence between the natural frequencies, only the modal coordinates corresponding to the respective natural frequencies are retained. The expressions for P and Q are as follows:

[0028] ;

[0029] In the formula, The difference in inertia of the asymmetric rotor in the x and y directions. Rotational speed;

[0030] The generalized displacement vector in the physical coordinate system Transformed to p in modal coordinates:

[0031] ;

[0032] in ;

[0033] In the formula, For along Axial translational degrees of freedom, For along Axial translational degrees of freedom, To bypass Rotational degrees of freedom of the axis To bypass The rotational degrees of freedom of the axis, a, b, c, d respectively correspond to , , , The degrees of freedom corresponding to each natural frequency in the modal coordinate system after transformation;

[0034] We obtain n=4 sets of decoupled second-order differential equations, where n is the number of degrees of freedom of the system:

[0035] .

[0036] Preferably, step S4 further includes:

[0037] Solve the second-order differential equation containing one degree of freedom 'a' after transformation and decoupling. The solution form for the system of differential equations containing the remaining degrees of freedom is the same, and its expression is:

[0038] ;

[0039] In the formula:

[0040] ;

[0041] make The expression is:

[0042] ;

[0043] in That is, the square of the first-order natural frequency, let:

[0044] ;

[0045] In the formula, δ represents the stability boundary. For perturbation;

[0046] Substituting it into the expression, we get:

[0047] ;

[0048] The power series expansion of the degrees of freedom a and the stability boundary δ is performed using the perturbation method:

[0049] ;

[0050] Where a(τ) has a period of π or 2π, substituting into the expression and comparing the perturbation term ε to the second-order term, we obtain three differential equations:

[0051] ;

[0052] ;

[0053] .

[0054] Preferably, step S5 further includes:

[0055] Solve the three differential equations in sequence. Solving the first differential equation yields a periodic solution of the form:

[0056] , ;

[0057] Discuss in turn By determining the stability boundary values, we can obtain the periodic solution of the inertial asymmetric rotor system response and the system's instability interval.

[0058] when When the periodic solution is:

[0059] ;

[0060] Substituting this into the second differential equation and eliminating the perennial term that tends to infinity with increasing time, we get Therefore, the particular solution is the zero solution. Similarly, substituting into the third differential equation, we get... ;

[0061] when When the periodic solution is:

[0062] ;

[0063] Substituting this into the second differential equation and eliminating the perennial term, we get:

[0064] ;

[0065] The special solution at this point is:

[0066] ;

[0067] Substituting this into the third differential equation and eliminating the perennial term, we get:

[0068] ;

[0069] Substitution ,get Two nearby stability boundaries;

[0070] The special solution at this point is:

[0071] ;

[0072] Substitution The rotational speed in the real modal space is equal to the natural frequency of that order. The amplitude response at time, through The response with actual amplitude is obtained by transforming back to the physical coordinate system;

[0073] when At this time, the periodic solution is:

[0074] ;

[0075] Substituting it into the second differential equation, without the perpetual term, we get ;

[0076] The special solution at this point is:

[0077] ;

[0078] Substituting this into the third differential equation and eliminating the perennial term, we get:

[0079] ;

[0080] Substitution ,get Two nearby stability boundaries;

[0081] The special solution at this point is:

[0082] ;

[0083] Substitution The rotational speed in the real modal space is equal to half of the natural frequency of that order. The amplitude response at time, through The response with actual amplitude is obtained by transforming back to the physical coordinate system;

[0084] With stability boundary The x-axis represents the difference in inertia. Using the vertical axis, in Plane, dividing line The perturbed zero solution has a period of or The periodic solution is the stability boundary of the system's zero solution; at the boundary line... Values ​​outside the interval have a stable zero solution, which is the stable interval; values ​​within the boundary interval have a disturbed zero solution that diverges exponentially, which is the unstable interval.

[0085] This application has the following beneficial effects:

[0086] The method of this application establishes a dynamic equation model of an inertial asymmetric rotor system; decomposes the dynamic equation of the inertial asymmetric rotor system, placing key influencing factor terms including rotational speed and inertia difference as parametric excitation terms on the right-hand side of the dynamic equation, considering the inertia difference as a disturbance term, and placing terms without parametric excitation terms on the left-hand side of the dynamic equation, thus writing the dynamic equation in matrix form; using real modal analysis, the left and right sides of the system dynamic equation are decoupled separately to obtain a decoupled second-order differential equation system; using the perturbation method, the degrees of freedom and stability boundaries in the decoupled differential equation system are expanded into power series and substituted into the differential equation system, listing the equation system according to the same power of the disturbance terms; solving the equation system sequentially, the periodic solution and instability interval of the inertial asymmetric rotor system response, as well as the analytical solution of the vibration response are obtained.

[0087] In this application, the key influencing factors of the stability of the inertial asymmetric rotor system are determined, namely the rotational speed and the difference in inertia in the directions of the two principal axes. Based on the key parameters of the asymmetric rotor, a simplified dynamic model is performed on a fixed coordinate system.

[0088] In this application, the key influencing factor terms including the difference between rotational speed and inertia are placed as parametric excitation terms on the right side of the dynamic equation, and the difference in inertia is considered as a disturbance term, that is, the rotational inertia of the system is considered to be time-varying.

[0089] In this application, the real modal analysis method is used to decouple the left and right ends of the system dynamic equations respectively, and obtain the decoupled second-order differential equation system, thereby improving the computational efficiency.

[0090] In this application, the perturbation method is used to expand the degrees of freedom and stability boundaries in the decoupled differential equation system into power series and substitute them into the differential equation system. The equation system is listed according to the same power of the perturbation terms, which has a unified iterative process and improves the calculation accuracy.

[0091] In this application, the equations are solved sequentially to obtain the periodic solution of the inertial asymmetric rotor system response and the system's instability interval. Considering the influence of rotational speed on the system's fundamental frequency, the analytical solution of the vibration response is obtained. Attached Figure Description

[0092] Figure 1 This is a flowchart illustrating a real modal analysis method for the stability of an inertial asymmetric rotor system according to an embodiment of this application.

[0093] Figure 2a , 2b 2c and 2c are schematic diagrams of the asymmetric rotor system in the x-axis, y-axis and z-axis directions, respectively, in the embodiments of this application.

[0094] Figure 3 This is a schematic diagram of the instability intervals corresponding to the first and second natural frequencies of the asymmetric rotor system in the embodiments of this application;

[0095] Figure 4 This is a schematic diagram of the instability range corresponding to the third and fourth natural frequencies of the asymmetric rotor system in the embodiments of this application;

[0096] Figure 5 This is a Campbell diagram of an asymmetric rotor system in the embodiments of this application;

[0097] Figure 6a and Figure 6b In the embodiments of this application, the asymmetric rotor system is... A schematic diagram of the stability boundaries corresponding to the first and second natural frequencies;

[0098] Figure 7a and Figure 7bThese are, respectively, the asymmetric rotor system in the embodiments of this application. A schematic diagram of the stability boundaries corresponding to the first and second natural frequencies. Detailed Implementation

[0099] To make the objectives, technical solutions, and advantages of this application clearer, the technical solutions in the embodiments of this application will be described in more detail below with reference to the accompanying drawings. In the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The described embodiments are only some, not all, of the embodiments of this application. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain this application, and should not be construed as limiting this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application. The embodiments of this application will be described in detail below with reference to the accompanying drawings.

[0100] This application provides a stability analysis method for an inertial asymmetric rotor system, such as... Figure 1 As shown, it mainly includes:

[0101] Step S1: Establish the dynamic equation model of the inertial asymmetric rotor system;

[0102] Step S2: Decompose the dynamic equation model of the inertial asymmetric rotor system, place the key influencing factor terms containing the speed and inertia difference as parameter excitation terms on the right side of the dynamic equation, consider the inertia difference as a disturbance term, place the terms without parameter excitation terms on the left side of the dynamic equation, and write the dynamic equation in matrix form.

[0103] Step S3: Using the real modal analysis method, the left and right sides of the matrix-form dynamic equations are decoupled respectively to obtain the decoupled second-order differential equation system.

[0104] Step S4: Use the perturbation method to expand the degrees of freedom and stability boundaries in the decoupled differential equation system into power series and substitute them into the differential equation system. List the equation system according to the same power of the perturbation terms.

[0105] Step S5: Solve the equations sequentially to obtain the periodic solution, instability interval, and analytical solution of the vibration response of the inertial asymmetric rotor system.

[0106] In some alternative implementations, step S1 further includes:

[0107] Based on the inertial asymmetric rotor system model, the dynamic equations of the inertial asymmetric rotor system are modeled in a fixed coordinate system.

[0108] In the case of neglecting damping, the dynamic equation of the inertial asymmetric rotor system is:

[0109] ;

[0110] In the formula, For the quality matrix, It is a gyroscope matrix. It is the stiffness matrix. It is a generalized displacement vector.

[0111] In this embodiment, based on the inertial asymmetric rotor system model, the dynamic equations of the inertial asymmetric rotor system are modeled in a fixed coordinate system. Four variables are defined in the fixed coordinate system: along... Axis translation ,along Axis translation , around Axis rotation , around Axis rotation ;

[0112] Because of the asymmetric distribution of inertia, in the body coordinate system of rotational speed Ω of direction and It has two different moments of inertia in the direction. and Angular velocity in a fixed coordinate system and Projected into the body coordinate system direction and Direction:

[0113] (1)

[0114] Projecting the system onto a fixed coordinate system, the kinetic energy equation is as follows:

[0115] (2)

[0116] In the formula, m is the rotor mass;

[0117] Projecting the system onto a fixed coordinate system, the potential energy equation is established as follows:

[0118] (3)

[0119] In the formula, , Let be the boundary stiffness of the left end of the asymmetric rotor system in the x and y directions. , Let be the boundary stiffness of the right end of the asymmetric rotor system in the x and y directions; a is the distance of the rotor from the left boundary in the system; b is the distance of the rotor from the right boundary in the system.

[0120] Applying Lagrange's energy theorem, we establish the system's energy equation:

[0121] (4)

[0122] In the formula, For the degrees of freedom of an asymmetric rotor, For the generalized forces corresponding to the degrees of freedom, the kinetic energy equation of the system is expanded as follows:

[0123] (5)

[0124] Expand the system's potential energy equation:

[0125] (6)

[0126] The rotor is simplified to a rectangular thin plate, therefore:

[0127] (7)

[0128] Assumption: Inertia difference The dynamic equations of the system obtained by solving this problem are:

[0129] (8)

[0130] In the formula, subscripts T, C, and R represent the translational stiffness coefficients at both ends of the asymmetric rotor system, the coupling stiffness coefficients between displacement and rotation, and the rotational stiffness coefficients, respectively:

[0131] , (9a)

[0132] , (9b)

[0133] , (9c)

[0134] Neglecting damping, the dynamic equations of the inertial asymmetric rotor system can be written in the form of standard dynamic equations:

[0135] (10)

[0136] In the formula, For the quality matrix, It is a gyroscope matrix. It is the stiffness matrix. It is a generalized displacement vector, expressed as:

[0137] (11a)

[0138] (11b)

[0139] (11c)

[0140] (11d)

[0141] In some alternative implementations, step S2 further includes:

[0142] By treating the key influencing factors of rotational speed and inertia difference as parametric excitation terms on the right-hand side of the dynamic equation, and considering the inertia difference as a disturbance term, while placing terms without parametric excitation terms on the left-hand side, the dynamic equation can be written in matrix form as follows:

[0143] (12)

[0144] In the formula, The mass matrix contains no parameterized excitation terms. It is a stiffness matrix that does not contain parametric excitation terms. It is a mass matrix containing parameter excitation terms. It is a gyroscope matrix containing parameter excitation terms.

[0145] In this embodiment, the matrix expressions are as follows:

[0146] (13a)

[0147] (13b)

[0148] (13c)

[0149] (13d)

[0150] In some alternative implementations, real modal analysis is used to decouple the left and right sides of the dynamic equations. The specific process is as follows:

[0151] Let the vector formed by the characteristic roots of the left-hand side of the dynamic equation be... , The eigenvector corresponding to each eigenvalue is ,in Arrange the eigenvectors into a 4x4 modality matrix. By decoupling the mass matrix M (without parameter excitation terms) and the stiffness matrix K (without parameter excitation terms) on the left-hand side of the dynamic equation using this modal matrix, the expression is obtained as follows:

[0152] (14)

[0153] In the formula, The modal principal mass matrix is... Given the modal principal stiffness matrix, the fundamental frequencies of the model are obtained by solving for the model's fundamental stiffness matrix.

[0154] The same transformation is applied to the mass matrix M' containing the parametric excitation term and the gyroscope matrix G' containing the parametric excitation term on the right-hand side of the dynamic equation, resulting in matrices P and Q, respectively:

[0155] (15)

[0156] Due to the independence between the natural frequencies, only the modal coordinates corresponding to the respective natural frequencies are retained. The expressions for P and Q are as follows:

[0157] (16)

[0158] In the formula, The difference in inertia of the asymmetric rotor in the x and y directions. Rotational speed;

[0159] The generalized displacement vector in the physical coordinate system Transformed to p in modal coordinates:

[0160] ;

[0161] in ;

[0162] In the formula, For along Axial translational degrees of freedom, For along Axial translational degrees of freedom, To bypass Rotational degrees of freedom of the axis To bypass The rotational degrees of freedom of the axis, a, b, c, d respectively correspond to , , , The degrees of freedom corresponding to each natural frequency in the modal coordinate system after transformation;

[0163] We obtain n=4 sets of decoupled second-order differential equations, where n is the number of degrees of freedom of the system:

[0164] (17)

[0165] In some optional implementations, in step S4, the perturbation method is used to perform a power series expansion of the degrees of freedom and stability boundaries in the decoupled differential equation system and substitute them into the differential equation system. The specific process includes:

[0166] Solve the second-order differential equation containing one degree of freedom 'a' after transformation and decoupling. The solution form for the system of differential equations containing the remaining degrees of freedom is the same, and its expression is:

[0167] (18)

[0168] In the formula:

[0169] (19)

[0170] make The expression is:

[0171] (20)

[0172] in That is, the square of the first-order natural frequency, let:

[0173] (twenty one)

[0174] In the formula, δ represents the stability boundary. For perturbation;

[0175] Substituting it into the expression, we get:

[0176] (twenty two)

[0177] The power series expansion of the degrees of freedom a and the stability boundary δ is performed using the perturbation method:

[0178] (twenty three)

[0179] Where a(τ) has a period of π or 2π, substituting into the expression and comparing the perturbation term ε to the second-order term, we obtain three differential equations:

[0180] (24a)

[0181] (24b)

[0182] (24c)

[0183] In some optional implementations, step S5 involves solving the system of differential equations sequentially to obtain the periodic solution of the inertial asymmetric rotor system response, the system's instability interval, and the analytical solution of the vibration response. Specifically, this includes:

[0184] Solve the three differential equations in sequence. Solving the first differential equation (24a) yields a periodic solution of the form:

[0185] , (25)

[0186] Discuss in turn By determining the stability boundary values, we can obtain the periodic solution of the inertial asymmetric rotor system response and the system's instability interval.

[0187] when When the periodic solution is:

[0188] (26)

[0189] Substituting this into the second differential equation (24b) and eliminating the perennial term that tends to infinity with increasing time, we get... Therefore, the particular solution is the zero solution. Similarly, substituting into the third differential equation (24c), we get... ;

[0190] when When the periodic solution is:

[0191] (27)

[0192] Substituting this into the second differential equation (24b), we get:

[0193]

[0194] (28)

[0195] The conditions for eliminating the perpetual age item are:

[0196] (29)

[0197] Solving for the given information, we get:

[0198] (30)

[0199] At this point, the particular solution of equation (28) is:

[0200] (31)

[0201] Substituting this into the third differential equation (24c), we get:

[0202]

[0203]

[0204]

[0205] (32)

[0206] The conditions for eliminating the perpetual age item are:

[0207]

[0208] (33)

[0209] Equation (33) can be expressed as:

[0210] (34)

[0211] in:

[0212] (35)

[0213] Solving for the given information, we get:

[0214] (36)

[0215] Substitution ,get Two nearby stable boundaries;

[0216] The special solution at this point is:

[0217] (37)

[0218] Substitution The rotational speed in the real modal space is equal to the natural frequency of that order. The amplitude response at time, through The response with actual amplitude is obtained by transforming back to the physical coordinate system;

[0219] when At this time, the periodic solution is:

[0220] (38)

[0221] Substituting this into the second differential equation (24b), we get:

[0222] (39)

[0223] No Yongnian Xiangde ,

[0224] At this point, the particular solution to equation (39) is:

[0225] (40)

[0226] Substituting this into the third differential equation (24c), we get:

[0227]

[0228] (41)

[0229] The conditions for eliminating the perpetual age item are:

[0230]

[0231] (42)

[0232] Equation (42) can be expressed as:

[0233] (43)

[0234] in:

[0235] (44)

[0236] Solving for the given information, we get:

[0237] (45)

[0238] Substitution ,get Two nearby stable boundaries;

[0239] The special solution at this point is:

[0240] (46)

[0241] Substitution The rotational speed in the real modal space is equal to half of the natural frequency of that order. The amplitude response at time, through The response with actual amplitude is obtained by transforming back to the physical coordinate system;

[0242] With stability boundary The x-axis represents the difference in inertia. Using the vertical axis, in Plane, dividing line The perturbed zero solution has a period of or The periodic solution is the stability boundary of the system's zero solution; at the boundary line... Values ​​outside the interval have a stable zero solution, which is the stable interval; values ​​within the boundary interval have a disturbed zero solution that diverges exponentially, which is the unstable interval.

[0243] In summary, the stability analysis method for an inertial asymmetric rotor system proposed in this application takes into account parametric excitation factors during calculation, which is more in line with actual working conditions. This application establishes a dynamic equation model for an inertial asymmetric rotor system. The dynamic equations of the inertial asymmetric rotor system are decomposed by placing key influencing factors, including the difference in rotational speed and inertia, as parametric excitation terms on the right-hand side of the equations. The inertia difference is considered as a disturbance term, indicating that the system's rotational inertia is time-varying. Terms without parametric excitation are placed on the left-hand side of the equations, resulting in a matrix form. The application employs real modal analysis to decouple the left and right sides of the dynamic equations, obtaining a decoupled set of second-order differential equations, thus improving computational efficiency. The application uses a perturbation method to expand the degrees of freedom and stability boundaries in the decoupled differential equations into power series and substitutes them into the equations. The equations are listed according to the same power of the disturbance terms, providing a unified iterative process and improving computational accuracy. The application solves the equations sequentially to obtain the periodic solution and instability interval of the inertial asymmetric rotor system response. Considering the influence of rotational speed on the system's fundamental frequency, an analytical solution for the vibration response is obtained.

[0244] Please see Figure 1 As shown in the embodiments of this application, a stability analysis method for an inertial asymmetric rotor system includes the following five steps:

[0245] 1) This embodiment of the application uses the asymmetric rotor model shown in Figure 2. The rotor's concentrated mass m = 2.13 kg, the distance from the rotor to the left end boundary a = 0.155 m, the distance from the rotor to the right end boundary b = 0.421 m, and the moment of inertia... Boundary stiffness of the left end of the model in the x-direction Boundary stiffness of the left end of the model in the y direction Boundary stiffness of the right end of the model in the x-direction Boundary stiffness of the right end of the model in the y direction ;

[0246] Based on the inertial asymmetric rotor system model, the dynamic equations of the inertial asymmetric rotor system are modeled in a fixed coordinate system. Four variables are defined in the fixed coordinate system, such as... Figure 2a , 2b As shown in 2c, they are respectively along Axis translation ,along Axis translation , around Axis rotation , around Axis rotation ;

[0247] The dynamic equations of the inertial asymmetric rotor system are established as shown in equation (10).

[0248] 2) Decompose the dynamic equations of the inertial asymmetric rotor system and identify the parametric excitation terms in the dynamic equations.

[0249] By treating the key influencing factors of rotational speed and inertia difference as parametric excitation terms on the right-hand side of the dynamic equation, and considering the inertia difference as a disturbance term, while placing terms without parametric excitation terms on the left-hand side, the dynamic equation can be written in matrix form as follows:

[0250] .

[0251] 3) The dynamic equations are decoupled using real modal analysis, resulting in n=4 sets of decoupled second-order differential equations, where n is the number of system degrees of freedom;

[0252] .

[0253] 4) The perturbation method is used to expand the degrees of freedom and stability boundaries in the decoupled differential equation system into power series and substitute them into the differential equation system to obtain three differential equations.

[0254] 5) Solve the system of differential equations sequentially to obtain the periodic solution of the inertial asymmetric rotor system response and the system's instability interval. For example... Figure 3 As shown, the instability regions corresponding to the first and second natural frequencies of the asymmetric rotor model are as follows: Figure 4 As shown, the instability regions corresponding to the third and fourth natural frequencies of the asymmetric rotor model are given. Figure 3 and Figure 4 It can be seen that the model stability boundary is Right now Time, and Right now hour. Figure 5 This is a Campbell plot for an asymmetric rotor model. The instability region is transformed onto the speed-inertia difference plane, and the fundamental frequency of the system is corrected based on the influence of the speed on the natural frequency. Figure 6 shows the instability region of the asymmetric rotor system. The stability boundaries corresponding to the first and second natural frequencies are shown in Figure 7 for the asymmetric rotor system. The stability boundaries corresponding to the first and second natural frequencies are shown in Figures 6 and 7. The marked points in Figures 6 and 7 are the model instability points obtained by the Runge-Kutta method, which are located within the instability interval calculated in this application, verifying the correctness of the calculation method in this application.

[0255] In summary, this application considers the system's rotational inertia to be time-varying, i.e., it considers the parametric excitation problem of an inertial asymmetric rotor. By using real modal analysis and perturbation methods, it obtains the analytical solution of the vibration response, acquires the instability range of the inertial asymmetric rotor system, and improves computational efficiency and accuracy.

[0256] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A stability analysis method for an inertial asymmetric rotor system, characterized in that, include: Step S1: Establish the dynamic equation model of the inertial asymmetric rotor system; Step S2: Decompose the dynamic equation model of the inertial asymmetric rotor system, place the key influencing factor terms containing the speed and inertia difference as parameter excitation terms on the right side of the dynamic equation, consider the inertia difference as a disturbance term, place the terms without parameter excitation terms on the left side of the dynamic equation, and write the dynamic equation in matrix form. Step S3: Using the real modal analysis method, the left and right sides of the matrix-form dynamic equations are decoupled respectively to obtain the decoupled second-order differential equation system. Step S4: Use the perturbation method to expand the degrees of freedom and stability boundaries in the decoupled differential equation system into power series and substitute them into the differential equation system. List the equation system according to the same power of the perturbation terms. Step S5: Solve the equations sequentially to obtain the periodic solution, instability interval, and analytical solution of the vibration response of the inertial asymmetric rotor system.

2. The stability analysis method for an inertial asymmetric rotor system according to claim 1, characterized in that, Step S1 further includes: Based on the inertial asymmetric rotor system model, the dynamic equations of the inertial asymmetric rotor system are modeled in a fixed coordinate system. In the case of neglecting damping, the dynamic equation of the inertial asymmetric rotor system is: ； In the formula, For the quality matrix, It is a gyroscope matrix. It is the stiffness matrix. It is a generalized displacement vector.

3. The stability analysis method for an inertial asymmetric rotor system according to claim 2, characterized in that, Step S2 further includes: By treating the key influencing factors of rotational speed and inertia difference as parametric excitation terms on the right-hand side of the dynamic equation, and considering the inertia difference as a disturbance term, while placing terms without parametric excitation terms on the left-hand side, the dynamic equation can be written in matrix form as follows: ； In the formula, The mass matrix contains no parameterized excitation terms. It is a stiffness matrix that does not contain parametric excitation terms. It is a mass matrix containing parameter excitation terms. It is a gyroscope matrix containing parameter excitation terms.

4. The stability analysis method for an inertial asymmetric rotor system according to claim 3, characterized in that, Step S3 further includes: Let the vector formed by the characteristic roots of the left-hand side of the dynamic equation be... , The eigenvector corresponding to each eigenvalue is ,in Arrange the eigenvectors into a 4x4 modality matrix. By decoupling the mass matrix M (without parameter excitation terms) and the stiffness matrix K (without parameter excitation terms) on the left-hand side of the dynamic equation using this modal matrix, the expression is obtained as follows: ； In the formula, The modal principal mass matrix is... Given the modal principal stiffness matrix, the fundamental frequencies of the model are obtained by solving for the model's fundamental stiffness matrix. The same transformation is applied to the mass matrix M' containing the parametric excitation term and the gyroscope matrix G' containing the parametric excitation term on the right-hand side of the dynamic equation, resulting in matrices P and Q, respectively: ； Due to the independence between the natural frequencies, only the modal coordinates corresponding to the respective natural frequencies are retained. The expressions for P and Q are as follows: ； In the formula, The difference in inertia of the asymmetric rotor in the x and y directions. Rotational speed; The generalized displacement vector in the physical coordinate system Transformed to p in modal coordinates: ； in ; In the formula, For along Axial translational degrees of freedom, For along Axial translational degrees of freedom, To bypass Rotational degrees of freedom of the axis To bypass The rotational degrees of freedom of the axis, a, b, c, d respectively correspond to , , , The degrees of freedom corresponding to each natural frequency in the modal coordinate system after transformation; We obtain n=4 sets of decoupled second-order differential equations, where n is the number of degrees of freedom of the system: 。 5. The stability analysis method for an inertial asymmetric rotor system according to claim 4, characterized in that, Step S4 further includes: Solve the second-order differential equation containing one degree of freedom 'a' after transformation and decoupling. The solution form for the system of differential equations containing the remaining degrees of freedom is the same, and its expression is: ； In the formula: ； make The expression is: ； in That is, the square of the first-order natural frequency, let: ； In the formula, δ represents the stability boundary. For perturbation; Substituting it into the expression, we get: ； The power series expansion of the degrees of freedom a and the stability boundary δ is performed using the perturbation method: ； Where a(τ) has a period of π or 2π, substituting into the expression and comparing the perturbation term ε to the second-order term, we obtain three differential equations: ； ； 。 6. The stability analysis method for an inertial asymmetric rotor system according to claim 5, characterized in that, Step S5 further includes: Solve the three differential equations in sequence. Solving the first differential equation yields a periodic solution of the form: ， ； Discuss in turn By determining the stability boundary values, we can obtain the periodic solution of the inertial asymmetric rotor system response and the system's instability interval. when When the periodic solution is: ； Substituting this into the second differential equation and eliminating the perennial term that tends to infinity with increasing time, we get Therefore, the particular solution is the zero solution. Similarly, substituting into the third differential equation, we get... ; when When the periodic solution is: ； Substituting this into the second differential equation and eliminating the perennial term, we get: ； The special solution at this point is: ； Substituting this into the third differential equation and eliminating the perennial term, we get: ； Substitution ,get Two nearby stability boundaries; The special solution at this point is: ； Substitution The rotational speed in the real modal space is equal to the natural frequency of that order. The amplitude response at time, through The response with actual amplitude is obtained by transforming back to the physical coordinate system; when At this time, the periodic solution is: ； Substituting it into the second differential equation, without the perpetual term, we get ; The special solution at this point is: ； Substituting this into the third differential equation and eliminating the perennial term, we get: ； Substitution ,get Two nearby stability boundaries; The special solution at this point is: ； Substitution The rotational speed in the real modal space is equal to half of the natural frequency of that order. The amplitude response at time, through The response with actual amplitude is obtained by transforming back to the physical coordinate system; With stability boundary The x-axis represents the difference in inertia. Using the vertical axis, in Plane, dividing line The perturbed zero solution has a period of or The periodic solution is the stability boundary of the system's zero solution; at the boundary line... Values ​​outside the interval have a stable zero solution, which is the stable interval; values ​​within the boundary interval have a disturbed zero solution that diverges exponentially, which is the unstable interval.

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