Rotor unbalance state characterization method comprehensively considering centroid offset and inertia main shaft inclination

Abstract

The invention discloses a rotor unbalance state characterization method comprehensively considering centroid offset and inertia main shaft inclination, and relates to the field of aero-engine rotor system unbalance state analysis, and the method comprises the steps: dividing each component in a rotor system into different structure units, and constructing a motion differential equation of a rotor based on the different structure units; applying rotary inertia load distribution excitation to the rotor by considering the mass center offset and inertia main shaft inclination state of each structural unit of the rotor; obtaining an explicit relationship between the rotor dynamic response and the rotor unbalance state based on work-energy balance; and comprehensive quantitative characterization of different positions and different forms of unbalance of the rotor is realized according to the work amount of the unbalance on the rotor. According to the method, the complex unbalance state of the rotor can be quantitatively represented through the lumped parameters, an analysis foundation is laid for follow-up rotor unbalance state control, and aero-engine rotor and whole machine vibration control is powerfully supported.

Description

Technical Field

[0001] The present invention relates to the field of unbalance state analysis of aero-engine rotor systems, and particularly to a method for characterizing the unbalance state of a rotor that comprehensively considers the centroid offset and the tilt of the inertia principal axis. Background Art

[0002] If the unbalance state of an aero-engine rotor is not good, excessive rotational excitation loads will be generated during its rotation, which will in turn lead to vibration problems of the rotor and even the entire engine. In severe cases, it will cause rotor-stator rubbing, bearing damage and failure, and structural fatigue failure, threatening the safe and reliable operation of the engine. Therefore, it is necessary to reasonably control the rotor unbalance state, which is an effective means to suppress the rotational inertia load of the rotor during rotation and relieve the vibration of the entire engine.

[0003] Due to the requirement for lightweight and high-efficiency structures in modern advanced aero-engines, while the rotor structural system continuously increases the rotational speed load, its bending stiffness is significantly reduced due to the lightweight design of the structure. Within the operating speed range, the rotor will inevitably undergo bending deformation and no longer meet the assumption of a steady-state rotor. That is, the unbalance state of the rotor will change with the operating speed. Correspondingly, the mechanical effects generated by the unbalance on the rotor vibration response will also change. The change in the rotor unbalance state and the mechanical effects generated by it due to the bending deformation are referred to as the change in the rotor structural state. Due to the change in the rotor structural state, it will have two aspects of influence on rotor unbalance control: on the one hand, due to the bending deformation of the rotor, there is a relative angular displacement between its different components, so the rotor and its unbalance state cannot be regarded as a whole, but the axial distribution characteristics of the unbalance of each component need to be considered; on the other hand, as the rotor undergoes bending deformation at high speeds, the components that make up it undergo angular tilt. In addition to considering their own mass eccentricity (the manifestation of component unbalance in the transverse degree of freedom), the influence of the tilt of the component inertia principal axis (the manifestation of component unbalance in the angular degree of freedom) also needs to be considered. In short, for the rotor structural system in advanced aero-engines, it is necessary to simultaneously consider the centroid offset and the tilt of the inertia principal axis of the components, and pay attention to the axial distribution characteristics of the unbalance state of each component.

[0004] However, in the existing rotor unbalance control requirements, based on the "steady-state rotor" assumption, the complex unbalance state of the rotor is often equivalently characterized by the transverse offset of the centroid of the entire rotor. Obviously, the traditional method for characterizing the rotor unbalance state does not consider the above-mentioned influence caused by rotor bending deformation and structural state change. Therefore, it is no longer applicable to the rotor system of modern advanced aero-engines and urgently needs to be developed. Summary of the Invention

[0005] In order to solve the problem that in the existing analysis of the influence of rotor unbalance distribution on the structural state, the additional skew of the inertial principal axis caused by the bending deformation of the high-speed rotor and the influence of the unbalance axial distribution characteristics brought by the relative angular deformation of different components are ignored, the present invention provides a method for characterizing the rotor unbalance state by comprehensively considering the centroid offset and the inclination of the inertial principal axis. For the rotational inertial force generated by the centroid offset of each component in the rotational state and the rotational inertial moment generated by the inclination of the inertial principal axis, through the lumped parameter of the total work done by the rotational inertial force / moment on the rotor, the quantitative characterization of the complex unbalance state of the rotor is realized, thereby laying an analysis foundation for the subsequent control of the rotor unbalance state and strongly supporting the vibration control of the aero-engine rotor and the whole machine.

[0006] To achieve the above object, the present invention adopts the following technical solutions:

[0007] A method for characterizing the rotor unbalance state by comprehensively considering the centroid offset and the inclination of the inertial principal axis, comprising the following steps:

[0008] Step 1: According to the stiffness / mass distribution of different components of the rotor, divide the rotor into several structural units, and obtain the mass , moment of inertia and bending stiffness of each structural unit, and establish the differential equation of motion of the rotor;

[0009] Step 2: Considering the centroid offset and the inclination state of the inertial principal axis of each structural unit of the rotor, apply a rotational inertial load distribution excitation to the rotor;

[0010] Step 3: Based on the work-energy balance, transform the differential equation of motion of the rotor under the rotational inertial load distribution excitation, and express the rotor deformation energy as an explicit expression of the centroid offset and the inclination of the inertial principal axis of each structural unit;

[0011] Step 4: According to the work done by the unbalance on the rotor, realize the comprehensive quantitative characterization of the unbalance at different positions and in different forms of the rotor.

[0012] Furthermore, quantitatively describe the unbalance state and the motion state of each structural unit of the rotor. Use the centroid offset and the inclination of the inertial principal axis of each structural unit to completely characterize the unbalance state of the rotor system; use the lateral motion and the angular motion of the centroid of each structural unit to completely characterize the motion state of the whole rotor.

[0013] Furthermore, establish the dynamic equation of the rotor system by the Lagrangian method, and based on the modal analysis method, solve the modal frequencies of the rotor near the working speed range and the displacement vectors of the corresponding vibration modes.

[0014] Furthermore, based on the unbalanced states of each structural unit of the rotor, a rotational inertia distribution excitation is applied to the rotor. In the rotating coordinate system, a mechanical model of the rotational inertia load distribution excitation of the rotor is established, and based on the characterization parameters of the unbalanced state and the motion state of the structural unit, the rotational inertia force / moment expressions related to the rotational speed are listed.

[0015] Furthermore, based on the work - energy balance, the differential equation of motion of the rotor under the rotational inertia load distribution excitation is transformed, and finally the deformation energy of the rotor is expressed as an explicit expression of the centroid offset and the tilt of the principal inertia axis of each structural unit. Taking the balance between the work of the rotational inertia load and the deformation energy of the rotor as the necessary and sufficient condition for the stable motion of the rotor, a mechanical model of the rotor motion based on the work of the inertial load of the structural unit and the elastic deformation energy of the rotor is established to jointly consider the influence of the rotational inertia load work generated by the centroid offset and the tilt of the principal inertia axis on the structural state of the rotor.

[0016] Furthermore, according to the work done by the unbalance on the rotor, a comprehensive quantitative characterization of different positions and different forms of unbalance of the rotor is realized.

[0017] Compared with the existing methods for characterizing the unbalanced state of the rotor, the present invention has the following beneficial effects:

[0018] The present invention aims at a rotor system with non - negligible bending deformation at high rotational speeds, considering the centroid offset and the tilt of the principal inertia axis of each component and their axial distribution characteristics, analyzes the influence of the rotor unbalance amount on the structural state and vibration of the rotor from the perspective of work - energy balance, and thus realizes the quantitative characterization of the complex unbalanced state of the rotor through lumped parameters, which is an important prerequisite for carrying out the optimization of the rotor unbalanced state, and thus strongly supports the vibration control of the aero - engine rotor and the whole machine. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 is a flowchart of the method for characterizing the unbalanced state of the rotor comprehensively considering the centroid offset and the tilt of the principal inertia axis of the present invention.

[0020] Figure 2a , Figure 2b is the distribution diagram of the rotational inertia load inside the rotor at different rotational speeds; among them, Figure 2a is the low - speed rotation region, Figure 2b is the high - speed rotation region.

[0021] Figure 3 is a schematic diagram of the mechanical model of the rotational inertia load distribution of the rotor. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0022] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0023] As Figure 1 shown, the present invention provides a method for characterizing the rotor unbalance state by comprehensively considering the centroid offset and the inclination of the principal axis of inertia, including the following steps:

[0024] Step 1: Divide the rotor into several structural units according to the stiffness / mass distribution of different components of the rotor. Among them, the compressor and turbine structural units are mass units, and the rotational inertial loads generated by their movements are the internal mechanical mechanisms of rotor deformation. Moreover, since the mass units have a large diameter and high stiffness, the bending deformation that occurs to themselves under working conditions can be ignored; structural units such as the front / rear journal, drum, and connection structure are elastic units and are prone to bending deformation when subjected to the rotational inertial loads generated by the mass units. Obtain the mass , moment of inertia (where represents the polar moment of inertia of the th structural unit, represents the diametral moment of inertia of the th structural unit), and bending stiffness to ensure that the established rotor dynamics model has the same stiffness / mass distribution characteristics as the actual rotor structure.

[0025] Establish an absolute coordinate system and a rotating coordinate system with the rotation center line as the reference. The absolute coordinate system is a coordinate system that is fixed and stationary in space , where axis is along the rotation center line direction of the rotor system, axis is perpendicular to the rotation center line direction. The establishment of this coordinate system is to describe the absolute position of points on the rotor in space. The rotating coordinate system is obtained by rotating the absolute coordinate system around the axis by a certain angle. The origin coordinates of the two coordinate systems are the same, axis coincides with the axis. The establishment of this coordinate system is to facilitate the analysis of rotor motion.

[0026] Based on the rotating coordinate system, establish a description method for the unbalance state and motion state of the rotor system to characterize the influence of the unbalance distribution of each structural unit on rotor deformation and motion when the rotor undergoes bending deformation. Specifically, with the centroid offset and the inclination of the principal axis of inertia Two parameters form the unbalance state characterization vector of the rotor to completely characterize the unbalance state of the rotor system, that is:

[0027] (1)

[0028] Among them, the superscript represents the th structural unit, and the subscripts and respectively represent the projections of the centroid offset / principal axis of inertia inclination vector in the and planes.

[0029] With the lateral displacement and angular displacement of the centroid of each structural unit as two parameters, form the motion displacement characterization vector of the rotor to completely characterize the overall motion state of the rotor, that is:

[0030] (2)

[0031] Among them, the superscript represents the th structural unit, and the subscripts and respectively represent the projections of the lateral / angular displacement vector in the and planes. The superscript T represents the transpose of the matrix.

[0032] Secondly, establish the differential equation of rotor motion based on the Lagrangian method. By to describe the position of the th structural unit in the absolute coordinate system, where and represent the th unit lateral displacement vector in the plane and plane projections, and respectively represent the structural unit angular displacement in the plane and plane projections. Based on this coordinate system, the kinetic energy of the structural unit can be expressed as:

[0033] (3)

[0034] Among them, 、 and are respectively the structural unit The mass, diameter moment of inertia, and polar moment of inertia, is the rotational speed of the rotor's self-rotation, is the structural unit at the velocity component in the plane, is the structural unit at the velocity component in the plane, is the structural unit at the angular velocity component in the plane, is the structural unit at the angular velocity component in the plane. Therefore, the strain energy between the th node and the th node can be obtained as:

[0035] (4)

[0036] where, and represent the stiffness matrix between the th node and the th node, and represent the relative displacement between nodes and .

[0037] Moreover, since the motion states of different structural units are significantly different after the high-speed rotor undergoes bending deformation, it is necessary to separately describe the lateral / angular displacements of different structural units within the rotor, and there is:

[0038] (5)

[0039] Based on the above analysis, the dynamic equation of the rotor is established by the Lagrangian method (6)

[0040] where, is the total derivative with respect to time , representing the rate of change of the parameter with respect to time; is the partial derivative with respect to the generalized velocity , represents the linear velocity or angular velocity of the th structural unit in the rotor system; represents the summation of the kinetic energies of all structural units in the system; represents the summation of the strain energies between all adjacent structural units in the system.

[0041] The mass matrix of the rotor structure system can be obtained , the damping matrix and the stiffness matrix , such that:

[0042] (7)

[0043] wherein, is the rotor displacement vector, is the rotor velocity vector, is the rotor acceleration vector. When the rotor performs synchronous forward precession at rotational speed, it can be assumed that has the form:

[0044] (8)

[0045] Based on this, the time term in Equation (7) can be differentiated to obtain:

[0046] (9)

[0047] And through coordinate transformation, it can be known that the rotor displacement vector in the rotating coordinate system satisfies:

[0048] (10)

[0049] wherein, represents the damping matrix in the rotating coordinate system, which is the transformation form of the damping matrix in the rotating coordinate system.

[0050] Based on the characteristic matrix in Equation (9), the natural mode vibration mode vector of the rotor at a specific rotational speed can be obtained as:

[0051] (11)

[0052] wherein, is the number of degrees of freedom of the rotor, is the resonance rotational speed of the rotor when the rotor rotational speed is , is the corresponding natural mode vibration mode vector, and then based on the modal analysis method, the displacement of the rotor in the rotating coordinate system can be expressed as a linear combination of the natural mode vibration mode vectors, and this displacement vector is only a function of the rotational speed , that is:

[0053] (12)

[0054] wherein, is the displacement vector of the rotor at a specific rotational speed under which the vector represents the displacement state of the rotor in the rotating coordinate system. Among them, there is , where is the weight coefficient of the th order modal vibration mode, and its physical meaning is that when the operating speed is , the severity of the th order modal deformation in the dynamic response of the rotor.

[0055] Step 2: Based on the unbalanced state of each structural unit of the rotor, by considering the centroid offset and the tilt state of the inertia principal axis of each structural unit of the rotor, a rotational inertia distribution excitation is applied to the rotor. Considering that there are centroid offsets and tilts of the inertia principal axes in each structural unit of the rotor, the rotational inertia loads generated during the rotor movement also include transverse rotational inertia forces and angular rotational inertia torques. As the rotational speed increases, the rotor undergoes bending deformation, and the distribution state of its rotational inertia load changes accordingly. Among them, as Figure 2a shown, when rotating at a low speed, the rotor does not undergo bending deformation, and the rotational inertia loads generated by each structural unit mainly act on the fulcrum positions, while the bending strain energy in the rotor shaft can be almost ignored. At this time, the rotational inertia load can be regarded as a concentrated load acting on the centroid position of the rotor , where is the rotational inertia force acting on the rotor, is the rotational inertia torque acting on the rotor, is the reaction force received by the front fulcrum, is the reaction force received by the rear fulcrum. As Figure 2b shown, when rotating at a high speed, the rotor undergoes bending deformation. At this time, high strain energy distributions will appear at the elastic unit and the connection structure positions, and the rotational inertia loads generated by the movement of each structural unit of the rotor can no longer be regarded as a whole, but the distribution characteristics of the loads need to be considered. Among them, is the rotational inertia force received by the compressor structural unit, is the rotational inertia torque received by the compressor structural unit, is the rotational inertia force received by the turbine structural unit, is the rotational inertia torque received by the turbine structural unit, is the reaction force received by the front fulcrum, is the reaction force received by the rear fulcrum.

[0056] Therefore, a mechanical model of the rotational inertia load distribution excitation of the rotor is established. As Figure 3 shown, the model considers the mass eccentricity of the structural unit and the tilt , and the local bending stiffness is corrected based on the virtual material method to characterize the local bending stiffness loss caused by the interface / geometric mutation discontinuity and its influence on the angular motion state of the structural unit.

[0057] Figure 3 In [Figure 1], the left side is a schematic diagram of a discontinuous rotor structure with interface connection and geometric mutation characteristics in the stationary state, and the right side is a local schematic diagram after the relative angular deformation of the structural unit in the high-speed rotation state. In the right figure, is the self-rotation speed of the rotor, is the precession speed of the rotor (i.e., the speed at which the rotor rotates around the rotation center line ), is the distance from the centroid of the structural unit to the rotation center line, is the distance from the center of mass of the structural unit to the centroid.

[0058] Based on the above characterization of the high-dimensional imbalance state and motion state of the structural unit, the rotational inertia force and the rotational inertia torque of the rotor structural unit can be expressed as:

[0059] (13)

[0060] where, , and are the mass and polar / diameter moment of inertia of the structural unit, respectively, which together with the initial centroid offset and the tilt of the principal axis of inertia characterize the imbalance state of the structural unit in the initial state; and are the lateral displacement and angular displacement of the structural unit, respectively, whose magnitudes vary with the operating speed , characterizing the motion state of the rotor. Through this expression, the influence of the rotor's rotational inertia load on the imbalance state and motion state of the rotor and its variation with the speed are established. Similarly, considering that the rotational inertia loads generated by different components at high speeds have a distribution characteristic, it is necessary to characterize the rotational inertia loads generated by each component separately, that is:

[0061] (14)

[0062] where the superscript represents the th structural unit, and the subscripts and represent the rotational inertia force / moment vector in and Projection in the plane. Substituting Equation (13) into Equation (14), an explicit expression of the rotational inertia load vector in terms of the rotor unbalance state and motion state is obtained, namely:

[0063] (15)

[0064] where, is the displacement vector of the inertia principal axes of each component of the rotor in the rotating coordinate system, and is the initial deflection vector of the inertia principal axes of each component of the rotor in the rotating coordinate system. diag represents a diagonal matrix.

[0065] To further simplify this expression, let:

[0066] (16)

[0067] Then Equation (15) is simplified to:

[0068] (17)

[0069] where, is the displacement vector of the inertia principal axes of each component of the rotor in the rotating coordinate system, and is the initial deflection vector of the inertia principal axes of each component of the rotor in the rotating coordinate system.

[0070] Since the matrix numerically satisfies:

[0071] (18)

[0072] Therefore, for Equation (10), we have:

[0073] (19)

[0074] indicating that in the rotating coordinate system, the rotational inertia load is balanced by the elastic restoring force generated by the rotor deformation.

[0075] Step 3: In the rotating coordinate system, due to the non-uniform dimensions of the rotational inertia force caused by the offset of the structural unit centroid and the rotational inertia moment caused by the inclination of the inertia principal axes, they cannot jointly describe the influence of both on the rotor dynamic response. Therefore, based on work-energy balance, the differential equation of motion of the rotor under the excitation of the rotational inertia load distribution is transformed to express the rotor deformation energy as an explicit expression of the offset of each structural unit centroid and the inclination of the inertia principal axes.

[0076] Based on Equations (14) and (17), the work W of the rotational inertia load generated by each structural unit during rotor motion can be expressed as:

[0077] (20)

[0078] When the displacement vector is , the deformation energy of the rotor is:

[0079] (21)

[0080] The balance between the work of the rotational inertia load and the deformation energy of the rotor is a necessary and sufficient condition for the rotor to achieve stable motion. Therefore, we have:

[0081] (22)

[0082] Thus, a rotor motion mechanics model based on the work of the inertial load of the structural unit and the elastic deformation energy of the rotor is established. Among them, the term ① on the right side of the equal sign represents the work done on the rotor due to the lateral / angular displacement of the structural unit when the rotor undergoes bending deformation, while the term ② represents the work done on the rotor due to the initial deviation of the inertial principal axis of the structural unit. Considering that the modal vibration mode vectors are orthogonal to the matrices and , we have:

[0083] (23)

[0084] Substitute Equation (12) into Equation (22), and at the same time eliminate the 0 term based on Equation (23), then the work of the inertial load can be decomposed based on the modal deformation, that is:

[0085] (24)

[0086] The left side of this equation is the deformation energy in the rotor when the th-order modal deformation occurs, while the right side is the work done by the rotational inertia load. And it should be noted that the work of the rotational inertia load includes two parts. Among them, the term ① represents the initial rotational inertia load work generated by the deviation of the inertial principal axis, while the term ② represents the additional rotational inertia load work generated during the additional deviation of the inertial principal axis relative to the rotational center axis when the rotor undergoes modal deformation.

[0087] Step 4: The rotational inertia excitation generated by the unbalanced distribution of the rotor is affected by the critical speed. Based on the work-energy conversion balance, solve the work-energy conversion process of the rotor under the rotational inertia excitation for different-order modal deformations.

[0088] Make a transformation on Equation to solve the weight coefficient of the th-order modal deformation, and we have:

[0089] (25)

[0090] And considering that there is: (26)

[0091] Further substituting Equation (24) and Equation (12) into Equation (23), we get:

[0092] (27)

[0093] According to this equation, it can be seen that the severity of the -th order modal deformation depends on three parts. The first part is the magnitude of the operating speed relative to the critical speed . When , the modal shape of this order theoretically reaches infinity. This is because near the critical speed, the kinetic energy and the strain potential energy in the rotor are almost equal ( ), and at this time the rotor is in a state of force balance, and external energy is more likely to be input into the rotor system. The second part is the inherent characteristic of the modal shape of this order. Due to the gyroscopic moment effect, this part will change with the operating speed . However, it can be predicted that when near the critical speed , the change degree of the value of the first part is much higher than that of the second part. The third part is the initial skew of the inertial principal axes of each component of the rotor, and further decomposing this part, we get:

[0094] (28)

[0095] On the right side of the equation, is the displacement component of the -th order modal shape vector in the degree of freedom . Among them, when is the lateral displacement degree of freedom, is multiplied by the initial offset of the component mass center. When is the angular degree of freedom, is multiplied by the initial inclination of the inertial principal axis. Obviously, is the influence weight coefficient of the initial unbalanced state of each component of the rotor on the rotor modal deformation. Numerically, it is equal to the displacement of the component at the corresponding degree of freedom when the rotor undergoes modal deformation. Its physical meaning is that when the rotor undergoes modal deformation under rotational inertia excitation, the more deformed the position is, the more work the inertial principal axis skew inputs the inertial load.

[0096] To sum up, with the change of the rotational speed, the rotational inertial load generated by the initial skew of the inertial principal axes of each component of the rotor will do work on the rotor, causing energy to converge in each order of modal deformation, and as the operating speed approaches the critical speed corresponding to the modal shape, the more easily energy is input into this mode, making the rotor deformation closer to the modal shape of this order.

[0097] As described above, it is only the specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any modification, equivalent replacement, improvement, etc. made by those skilled in the art within the technical scope disclosed by the present invention shall be covered by the protection scope of the present invention.

Claims

1. A method for characterizing the unbalanced state of a rotor by comprehensively considering the centroid offset and the tilt of the principal inertia axis, characterized in that It includes the following steps: Step 1: Divide the rotor into several structural units according to the stiffness / mass distribution of different components of the rotor, and obtain the mass of each structural unit , moment of inertia and bending stiffness , and establish the differential equation of motion of the rotor; Step 2: Considering the centroid offset and the inclination state of the principal axis of inertia of each structural unit of the rotor, apply a rotational inertia load distribution excitation to the rotor; Step 3: Based on the work - energy balance, transform the differential equation of motion of the rotor under the rotational inertia load distribution excitation, and express the rotor deformation energy as an explicit expression of the centroid offset and the inclination of the principal axis of inertia of each structural unit; Step 4: According to the work done by the imbalance on the rotor, achieve a comprehensive quantitative characterization of different positions and different forms of imbalance of the rotor.

2. The rotor unbalance state characterization method considering the centroid offset and the tilt of the inertia principal axis according to claim 1, wherein The said Step 1 includes: Using the centroid offset of each structural unit and the inclination of the principal axis of inertia These two parameters form an unbalance state characterization vector to fully characterize the unbalance state of the rotor system, that is: （1） Among them, the superscript represents the th structural unit, and the subscripts and represent the projections of the centroid offset / principal inertia axis tilt vector in the and planes in the rotating coordinate system; the superscript T represents the transpose of the matrix; With the lateral displacements of the centroids of each structural unit and the angular displacements Two parameters are used to form a motion displacement characterization vector to fully characterize the motion state of the rotor system, that is: （2） Among them, the superscript indicates the th structural unit, and the subscripts and represent the projections of the transverse / angulardisplacement vectors in the and planes; Through to describe the position of the th structural unit in the absolute coordinate system, where and represent the th unit lateral displacement vector in the plane and plane projection, and represent the angular displacement of the structural unit in the plane and plane projection; based on this coordinate system, the kinetic energy of the structural unit is expressed as: ​ （3） Among them, , and are the mass, diameter moment of inertia, and polar moment of inertia of the structural unit , respectively. is the self-rotation speed of the rotor. is the velocity component of the structural unit in the plane. is the velocity component of the structural unit in the plane. is the angular velocity component of the structural unit in the plane. is the angular velocity component of the structural unit in the plane; thus, the strain energy between the th node and the is obtained as follows: （4） Among them, and represent the stiffness matrix between the -th node and the -th node, and represent the relative displacement between nodes and ; Lateral / Angular Displacements of Different Structural Units within the Rotor are separately described, and there is: （5） Establish the dynamic equation of the rotor by the Lagrangian method as: （6） Among them, is the total derivative with respect to time , representing the rate of change of the parameter with respect to time; is the partial derivative with respect to the generalized velocity , representing the linear velocity or angular velocity of the th structural unit in the rotor system; represents the summation of the kinetic energies of all structural units in the system; represents the summation of the deformation energies between all adjacent structural units in the system; Obtain the mass matrix of the rotor structure system , damping matrix and stiffness matrix , such that: （7） Among them, is the rotor displacement vector, is the rotor velocity vector, is the rotor acceleration vector; When the rotor makes synchronous forward precession at the rotational speed, assume that has the form: （8） Based on this, the time term in the dynamic equation can be differentiated to obtain: （9） And through coordinate transformation, it can be known that the rotor displacement vector in the rotating coordinate system satisfies: （10） Among them, represents the damping matrix in the rotating coordinate system, which is the transformation form of the damping matrix in the rotating coordinate system; Obtain the natural mode vibration shape vector of the rotor at a specific rotational speed as: （11） Among them, is the number of degrees of freedom of the rotor, is the rotor speed at when the resonance speed of the rotor corresponding to the natural mode vibration shape vector. Subsequently, based on the modal analysis method, the displacement of the rotor in the rotating coordinate system is expressed as a linear combination of the natural mode vibration shape vectors, and this displacement vector is only a function of the rotational speed , that is: （12） Among them, is the displacement vector of the rotor at a specific rotational speed , and this vector represents the displacement state of the rotor in the rotating coordinate system; among them, there is , where is the weight coefficient of the -th order modal vibration mode, and its physical meaning is that when the operating speed is , the severity of the -th order modal deformation in the dynamic response of the rotor.

3. The rotor unbalance state characterization method considering the centroid offset and the tilt of the inertia principal axis according to claim 1, characterized in that The said Step 2 includes: The mechanical model of the excitation of the rotor's rotational inertia load distribution considers the mass eccentricity of the structural unit and the inclination of the principal axis of inertia , and modifies the local bending stiffness based on the virtual material method to characterize the local bending stiffness loss caused by the interface / geometric mutation discontinuity and its influence on the angular motion state of the structural unit.

4. The method for characterizing the rotor unbalance state considering the centroid offset and the inclination of the principal inertia axis according to claim 3, wherein Based on the above characterization of the high-dimensional imbalance state and motion state of the structural unit, the rotational inertia force of the rotor structural unit , and the rotational inertia torque are expressed as: （13） Among them, , and are the mass of the structural unit and the polar / diameter moment of inertia respectively, and together with the initial centroid offset of the structural unit and the tilt of the principal axis of inertia collectively characterize the unbalanced state of the structural unit in the initial state; and are the lateral displacement and angular displacement of the structural unit respectively, and their magnitudes vary with the operating speed and characterize the motion state of the rotor; through this expression, the influence of the rotational inertial load of the rotor on the unbalanced state and motion state of the rotor, and its variation with the rotational speed are established.

5. The method for characterizing the rotor unbalance state considering the centroid offset and the inclination of the principal axis of inertia according to claim 4, wherein Considering that the rotational inertia loads generated by different components at high rotational speeds have a distribution characteristic, it is necessary to respectively characterize the rotational inertia loads generated by each component, that is: （14） Among them, the superscript represents the th structural unit, and the subscripts and represent the projections of the rotational inertia force / moment vector in the and planes respectively; substituting the rotational inertia load vector into the above formula, an explicit expression of the rotational inertia load vector with respect to the rotor unbalance state and motion state can be obtained, that is: （15） Among them, is the displacement vector of the inertia principal axis of each rotor component in the rotating coordinate system, while is the initial skew vector of the inertia principal axis of each rotor component in the rotating coordinate system, and diag represents a diagonal matrix; to further simplify this expression, let (16) Thus, it is simplified to: （17） Since the matrix numerically satisfies (18) Therefore, there is (19).

6. The method for characterizing the rotor unbalance state considering the centroid offset and the inclination of the inertia principal axis according to claim 1, wherein The said Step 3 includes: The work done by the rotational inertia load generated by each structural unit during the rotor motion is expressed as: （20）。 7. The method for characterizing the rotor unbalance state considering the centroid offset and the inclination of the principal axis of inertia according to claim 6, wherein When the displacement vector is the deformation energy of the rotor is: （21） The balance of the rotational inertial load work and the rotor deformation energy is a necessary and sufficient condition for the rotor to achieve stable motion. Therefore, there is (22) Thereby establish a rotor motion mechanics model based on the inertial load work of the structural unit and the elastic deformation energy of the rotor, where term ① on the right side of the equal sign represents the work done on the rotor due to the lateral / angular displacement of the structural unit when the rotor undergoes bending deformation, and term ② represents the work done on the rotor due to the initial skew of the principal axis of inertia of the structural unit; Considering the orthogonality of the modal vibration mode vectors with respect to the matrix and we have: （23）。 8. The method for characterizing the rotor unbalance state considering the centroid offset and the tilt of the inertia principal axis according to claim 7, wherein Substitute into , and at the same time, based on the matrices and , eliminate the zero terms orthogonally. Then, decompose the inertial load work based on the modal deformation, i.e.: （24） On the left side of this equation is the deformation energy in the rotor when the -order modal deformation occurs, and on the right side is the work done by the rotational inertia load. The work done by the rotational inertia load consists of two parts. Among them, item ① represents the initial rotational inertia load work generated by the skew of the inertia principal axis, and item ② represents the additional rotational inertia load work generated during the additional skew of the inertia principal axis relative to the rotation center axis when the rotor undergoes modal deformation.

9. The method for characterizing the rotor unbalance state comprehensively considering the centroid offset and the inclination of the principal axis of inertia according to claim 1, characterized in that The said Step 4 includes: For perform a transformation to solve for the weight coefficients of the -th order modal deformation, we have: ​ （25） Considering that there are (26) Then there is: （27）。 10. The method for characterizing the rotor unbalance state considering the centroid offset and the inclination of the inertia principal axis according to claim 9, wherein Obtained according to Equation (27), the severity of the modal deformation of the nth order depends on three parts. The first part is the magnitude of the operating speed relative to the critical speed , the second part is the inherent characteristics of the modal shape of this order, and the third part is the initial deviation of the inertia principal axis of each component of the rotor. Further decomposition of this part gives: （28） On the right side of the equation is the -th order modal vibration mode vector displacement component at the degree of freedom . Among them, when is the lateral displacement degree of freedom, is multiplied by the initial offset of the component centroid . When is the angular degree of freedom, is multiplied by the initial inclination of the inertia principal axis . is the influence weight coefficient of the initial unbalance state of each component of the rotor on the modal deformation of the rotor. Numerically, it is equal to the displacement of the component at the corresponding degree of freedom when the rotor undergoes modal deformation. Its physical meaning is that when the rotor undergoes modal deformation under rotational inertia excitation, the greater the deformation, the more work is input by the inertial load with the skew of the inertia principal axis at that position.

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