WO2023213017A1 - Rotor system dynamic characteristic calculation method and device based on finite element method, and medium - Google Patents

Rotor system dynamic characteristic calculation method and device based on finite element method, and medium Download PDF

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WO2023213017A1
WO2023213017A1 PCT/CN2022/108728 CN2022108728W WO2023213017A1 WO 2023213017 A1 WO2023213017 A1 WO 2023213017A1 CN 2022108728 W CN2022108728 W CN 2022108728W WO 2023213017 A1 WO2023213017 A1 WO 2023213017A1
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disk
rotor system
equation
shaft segment
motion
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PCT/CN2022/108728
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French (fr)
Chinese (zh)
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杜飞平
胡宝文
谭永华
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西安航天动力研究所
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

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  • This application relates to the technical field of rotor system dynamics systems, and in particular to methods, equipment and media for calculating dynamic characteristics of rotor systems based on the finite element method.
  • the research object of rotor system dynamics is a rotor whose lateral displacement is much smaller than the shaft diameter (on the order of 0.1%). Its vibration includes various forms such as torsional vibration and bending vibration of the rotating shaft, disc vibration or disc jitter, among which the bending vibration of the rotating shaft is It is the most complex and involves the most factors. Therefore, the rotor system dynamics takes the transverse bending vibration of the rotating shaft as the main research object.
  • the calculation methods of modern rotor system dynamics can be divided into transfer matrix method and finite element method:
  • the transfer matrix method is characterized by the fact that the matrix order does not increase with the increase of the system degrees of freedom, so it has simple programming, small memory and fast operation speed.
  • this application discloses a method, equipment and medium for calculating the dynamic characteristics of a rotor system based on the finite element method, which is used to provide a process and numerical algorithm for solving the rotor characteristics using the finite element method.
  • this application provides a method for calculating the dynamic characteristics of a rotor system based on the finite element method, which is characterized in that the rotor system includes a disc, a bearing, and an elastic shaft connecting the disc and the bearing; the method is based on the finite element method.
  • the calculation methods of the dynamic characteristics of the rotor system using the finite element method include:
  • this application first calculates the motion equation of the disk based on the Euler angle rotation transformation matrix, calculates the motion equation of the elastic shaft based on the displacement interpolation function matrix, and calculates the bearing journal center coordinates based on the The equation of motion of the rotor system is then determined based on the equation of motion of the disk, the equation of motion of the elastic shaft and the equation of motion of the bearing. Finally, based on the equation of motion of the rotor system, the critical rotational speed and inconsistency of the rotor system are calculated. Balanced response.
  • the method for calculating the dynamic characteristics of the rotor system based on the finite element method can use the finite element method to solve the equations of motion during the rotation of the rotor system and calculate the critical speed and unbalance response of the rotor system.
  • embodiments of the present application provide a device for calculating dynamic characteristics of a rotor system based on the finite element method, including a processor and a communication interface coupled to the processor; the processor is used to run computer programs or instructions to implement calculations based on Finite element method is used to calculate the dynamic characteristics of the rotor system.
  • embodiments of the present application provide a computer storage medium. Instructions are stored in the computer storage medium. When the instructions are executed, a method for calculating the dynamic characteristics of the rotor system based on the finite element method is implemented.
  • the second aspect and the third aspect of the present application have the same beneficial effects as the above-mentioned technical solution assessment and evaluation method, and will not be described again here.
  • Figure 1 is a step flow chart of a method for calculating dynamic characteristics of a rotor system based on the finite element method provided by an embodiment of the present application;
  • Figure 2 is a schematic diagram of a rotor system supported by a rolling bearing provided in an embodiment of the present application
  • FIG. 3 is a schematic diagram of a shaft segment unit provided by an embodiment of the present application.
  • Figure 4 is a schematic diagram of the hardware structure of a rotor system dynamic characteristics calculation device based on the finite element method provided by an embodiment of the present application;
  • FIG. 5 is a schematic structural diagram of a chip provided by an embodiment of the present application.
  • FIG. 1 shows a step flow chart of a method for calculating dynamic characteristics of a rotor system based on the finite element method provided by an embodiment of the present application.
  • the rotor system includes a disc, a bearing and an elastic shaft connecting the disc and the bearing, where the bearing in Figure 2 is a rolling bearing.
  • the bearing in Figure 2 is a rolling bearing.
  • l is the length of the elastic shaft
  • O 1 and O 3 are the journal centers at the rolling bearing support
  • O 2 is the rotor center at the disk.
  • the position of the disk's rotating shaft at any cross-section can be determined by the axis coordinates x, y , section rotation angle ⁇ x , ⁇ y and rotation angle to calculate.
  • the above-mentioned calculation method of dynamic characteristics of the rotor system based on the finite element method includes the following steps:
  • x and y respectively represent the abscissa and ordinate of the axis of the disk, represent the first-order reciprocal of the abscissa and ordinate of the axis of the disk to time respectively
  • m d , J d and J p are the mass, diameter moment of inertia and polar moment of inertia of the rigid disc respectively
  • O 2 ⁇ is The axis node is the origin, and the O 2 ⁇ axis is perpendicular to the disk plane and is fixed in the moving coordinate system of the disk.
  • ⁇ ⁇ , ⁇ ⁇ , and ⁇ ⁇ respectively represent the first component and the second component of the rotational speed in the moving coordinate system. component and third component.
  • Equation (6-2) omitting the second-order and above traces, the motion equation of the disk can be obtained as:
  • M d represents the generalized mass of the disk
  • ⁇ represents the rotation angular velocity of the disk
  • J represents the rotational inertia of the disk.
  • T is the kinetic energy of the system, q i , are generalized displacement and generalized velocity respectively.
  • ⁇ Q 1d ⁇ [F x , M y ] T
  • ⁇ Q 2d ⁇ [F y , M x ] T
  • Q 1d and Q 2d respectively represent the normal contact load
  • F x and F y respectively represent the first and second components of the external force on the disk
  • M x and M y respectively represent the first component of the moment. and the second component.
  • This step specifically includes: S201, dividing the elastic shaft into multiple shaft segment units.
  • FIG. 3 shows a schematic structural diagram of the shaft segment unit.
  • the generalized coordinates of the shaft segment unit are the displacements of the two nodes, that is,
  • x A , x B , y A , y B , ⁇ xA , ⁇ xB , ⁇ yA , ⁇ yB respectively represent the coordinates of point A and point B along the X direction, the coordinates of the Y direction, the angle between the X direction, Y direction angle.
  • S202 Calculate the kinetic energy and bending potential energy of the shaft segment unit using the displacement interpolation function matrix and the node displacement of the shaft segment unit.
  • the displacement at any section of the shaft segment unit can be calculated through the displacement interpolation function and the displacement of the node of the shaft segment unit.
  • [N] [N 1 (z) N 2 (z) N 3 (z) N 4 (z)] is a 1 ⁇ 4 order displacement interpolation function matrix.
  • the displacement interpolation function can be solved as
  • x (z, t), y (z, t) respectively represent the displacement at any section of the shaft segment unit
  • ⁇ x (z, t), ⁇ y (z, t) represent the shaft segment unit
  • the angular displacement at any section, u 1z and u 2z respectively represent the third component of the disk rotation in the Cartesian coordinate system, z represents the position, t represents the time, and N is the displacement interpolation function;
  • the displacement of any point of the axis segment unit can be expressed by the displacement of the node of the unit, and the kinetic energy and potential energy of the unit can also be expressed as functions of the node displacement and velocity.
  • the kinetic energy of the shaft segment micro-element can be obtained:
  • dm, j d and j p respectively represent the mass, diameter moment of inertia and polar moment of inertia of the shaft segment element.
  • dm, j d and j p respectively represent the mass, diameter moment of inertia and polar moment of inertia of the shaft segment unit, and N' represents the quadratic displacement interpolation polynomial of the shaft segment unit.
  • E is the elastic modulus
  • I is the moment of inertia of the cross section against the bending neutral axis
  • M represents the third-order displacement interpolation polynomial of the axis segment unit, and Respectively represent the acceleration along the x and y directions at any section of the shaft segment unit.
  • M zT and M zR respectively represent the diagonal mass matrix representing the axis segment unit, and J z represents the moment of inertia of the axis segment unit;
  • K z represents the stiffness matrix of the axis segment element.
  • ⁇ Q 1z ⁇ , ⁇ Q 2z ⁇ are the corresponding generalized force vectors, including the disks or adjacent shaft segments connected at the nodes. forces, moments and unbalanced forces.
  • the unbalanced force of the shaft segment element can be obtained as:
  • e ⁇ (z) and e ⁇ (z) are the eccentricity of the mass distribution of the shaft segment unit.
  • x b and y b are the coordinates of the bearing seat center of the bearing
  • x s(j) and y s(j) are the coordinates of the journal center of the bearing
  • c bxx , c bxy , c byx , c byy respectively represent the damping of the bearing in different directions.
  • ⁇ U 1 ⁇ [x 1 , ⁇ y1 , x 2 , ⁇ y2 ,..., x n , ⁇ yn ] T
  • [M], ⁇ [J], and [K] are the mass matrix, rotation matrix, and stiffness matrix of the rotor system respectively
  • ⁇ Q 1 ⁇ , ⁇ Q 2 ⁇ are the generalized forces of the rotor system.
  • Equation (6-25) can be written in the unified form as
  • the critical speed of the rotor system refers to the speed when the rotor system resonates under its own unbalanced excitation force.
  • the critical speed characteristics are the inherent characteristics of the rotor system.
  • the turbine frequency when the rotational angular velocity is ⁇ can be obtained through the homogeneous solution of the differential equation.
  • [M] and [K ⁇ are real symmetric matrices
  • [J] is a real antisymmetric matrix
  • the eigenvalues of this equation are composed of 4n pairs of conjugate complex numbers, corresponding to 2n forward whirl frequencies and 2n reverse whirl frequencies respectively.
  • unbalanced response The vibration generated by the rotor system under the excitation of unbalanced force or unbalanced torque is called unbalanced response, which is mainly used to study the sensitivity of the rotor system to the unbalanced amount at certain positions.
  • ⁇ z ⁇ ⁇ U 1 ⁇ + ⁇ U 2 ⁇
  • i represents the imaginary number
  • Q 1c and Q 2c represent the unbalanced force respectively.
  • the embodiment of the present application first calculates the motion equation of the disk based on the Euler angle rotation transformation matrix, calculates the motion equation of the elastic shaft based on the displacement interpolation function matrix, and calculates the motion equation of the bearing based on the journal center coordinates of the bearing. , then, determine the motion equation of the rotor system based on the motion equation of the disk, the elastic shaft, and the bearing. Finally, based on the motion equation of the rotor system, calculate the critical speed and unbalance response of the rotor system.
  • the method for calculating the dynamic characteristics of the rotor system based on the finite element method can use the finite element method to solve the equations of motion during the rotation of the rotor system and calculate the critical speed and unbalanced response of the rotor system.
  • Figure 4 shows a schematic diagram of the hardware structure of a device for calculating dynamic characteristics of a rotor system based on the finite element method provided by an embodiment of the present application.
  • the rotor system dynamic characteristics calculation device 80 based on the finite element method includes a processor 801 and a communication interface 802 .
  • the above-mentioned processor can be a general central processing unit (CPU), a microprocessor, an application-specific integrated circuit (ASIC), or one or more processors for controlling the computer.
  • the application program program is implemented on an integrated circuit.
  • the communication interface may use any device such as a transceiver for communicating with other devices or communication networks.
  • the above-mentioned finite element method-based rotor system dynamic characteristics calculation device may also include a communication line 803 .
  • the communication line may include a path to carry information between the above-mentioned components.
  • the finite element method-based rotor system dynamic characteristics calculation device may also include a memory 804 .
  • the memory is used to store computer execution instructions for executing the solution of the present application, and is controlled by the processor for execution.
  • the processor is used to execute computer execution instructions stored in the memory, thereby implementing the method provided by the embodiment of the present application.
  • the above-mentioned memory can be a read-only memory (ROM) or other types of static storage devices that can store static information and instructions, a random access memory (random access memory, RAM) or a Other types of dynamic storage devices for information and instructions, which may also be electrically erasable programmable read-only memory (EEPROM), compact disc read-only memory (CD-ROM) or Other optical disc storage, optical disc storage (including compressed optical discs, laser discs, optical discs, digital versatile discs, Blu-ray discs, etc.), magnetic disk storage media or other magnetic storage devices, or can be used to carry or store desired information in the form of instructions or data structures Program code and any other medium capable of being accessed by a computer, without limitation.
  • the memory can exist independently and be connected to the processor through communication lines. Memory can also be integrated with the processor.
  • the computer-executed instructions in the embodiments of the present application may also be called application codes, which are not specifically limited in the embodiments of the present application.
  • the processor 801 may include one or more CPUs, such as CPU0 and CPU1 in FIG. 4 .
  • the rotor system dynamic characteristics calculation device based on the finite element method may include multiple processors, such as processor 801-1 and processor 801- in Figure 4 2. Each of these processors can be a single-core processor or a multi-core processor.
  • FIG. 5 is a schematic structural diagram of a chip provided by an embodiment of the present application.
  • the chip 90 includes one or more (including two) processors 801 and a communication interface 802 .
  • the chip also includes a memory 804, which can include read-only memory and random access memory, and provides operating instructions and data to the processor.
  • a memory 804 can include read-only memory and random access memory, and provides operating instructions and data to the processor.
  • Part of the memory may also include non-volatile random access memory (NVRAM).
  • NVRAM non-volatile random access memory
  • the memory stores the following elements, execution modules or data structures, or their subsets, or their extended sets.
  • the corresponding operation is performed by calling the operation instructions stored in the memory (the operation instructions can be stored in the operating system).
  • the processor controls the processing operations of any one of the rotor system dynamic characteristics calculation devices based on the finite element method.
  • the processor can also be called a central processing unit (CPU).
  • memory may include read-only memory and random access memory and provide instructions and data to the processor.
  • Part of the memory may also include NVRAM.
  • the memory, communication interface and memory are coupled together through a bus system.
  • the bus system may also include a power bus, a control bus, a status signal bus, etc.
  • the various buses are labeled bus system 805 in Figure 5.
  • the method disclosed in the above embodiment of the present application can be applied in a processor or implemented by the processor.
  • the processor may be an integrated circuit chip that has signal processing capabilities.
  • each step of the above method can be completed by instructions in the form of hardware integrated logic circuits or software in the processor.
  • the above-mentioned processor can be a general-purpose processor, digital signal processing (DSP), ASIC, off-the-shelf programmable gate array (field-programmable gate array, FPGA) or other programmable logic devices, discrete gates or transistor logic. devices, discrete hardware components.
  • DSP digital signal processing
  • ASIC application-the-shelf programmable gate array
  • FPGA field-programmable gate array
  • Each method, step and logical block diagram disclosed in the embodiment of this application can be implemented or executed.
  • a general-purpose processor may be a microprocessor or the processor may be any conventional processor, etc.
  • the steps of the method disclosed in conjunction with the embodiments of the present application can be directly implemented by a hardware decoding processor, or executed by a combination of hardware and software modules in the decoding processor.
  • the software module can be located in random access memory, flash memory, read-only memory, programmable read-only memory or electrically erasable programmable memory, registers and other mature storage media in this field.
  • the storage medium is located in the memory, and the processor reads the information in the memory and completes the steps of the above method in combination with its hardware.
  • the communication interface is used to obtain images collected by the camera.
  • the processor is used to execute steps 101 to 103 of the assessment and evaluation method in the embodiment shown in FIG. 1 .
  • a computer-readable storage medium is provided. Instructions are stored in the computer-readable storage medium. When the instructions are executed, the functions performed by the rotor system dynamic characteristics calculation device based on the finite element method in the above embodiments are realized.
  • a chip is provided.
  • the chip is used in a rotor system dynamic characteristics calculation device based on the finite element method.
  • the chip includes at least one processor and a communication interface.
  • the communication interface is coupled with at least one processor.
  • the processor is used to run instructions. In order to realize the functions performed by the rotor system dynamic characteristics calculation device based on the finite element method in the above embodiments.
  • the computer program product includes one or more computer programs or instructions.
  • the computer may be a general purpose computer, a special purpose computer, a computer network, a terminal, a user equipment, or other programmable device.
  • the computer program or instructions may be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another.
  • the computer program or instructions may be transmitted from a website, computer, A server or data center transmits via wired or wireless means to another website site, computer, server, or data center.
  • the computer-readable storage medium may be any available medium that can be accessed by a computer or a data storage device such as a server or data center that integrates one or more available media.
  • the available media may be magnetic media, such as floppy disks, hard disks, and magnetic tapes; they may also be optical media, such as digital video discs (DVDs); they may also be semiconductor media, such as solid state drives (solid state drives). ,SSD).

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Abstract

The present application relates to the technical field of rotor system dynamics, discloses a rotor system dynamic characteristic calculation method and device based on a finite element method, and a medium, and is used for providing a process and numerical algorithm for solving rotor characteristics by using the finite element method. The rotor system dynamic characteristic calculation method based on the finite element method comprises: calculating a motion equation of the disc on the basis of an Euler angle rotation transformation matrix; calculating a motion equation of the elastic shaft by utilizing a displacement interpolation function matrix; calculating a motion equation of a bearing on the basis of a journal center coordinate of the bearing; determining a motion equation of the rotor system on the basis of the motion equation of the disc, the motion equation of the elastic shaft, and the motion equation of the bearing; and calculating a critical rotating speed and an unbalance response of the rotor system on the basis of the motion equation of the rotor system.

Description

一种基于有限元法的转子系统动特性计算方法、设备及介质A method, equipment and medium for calculating the dynamic characteristics of a rotor system based on the finite element method
本申请要求于2022年05月05日提交中国专利局、申请号为202210481719.6、申请名称为“基于有限元法的转子系统动特性计算方法、设备及介质”的中国专利优先权，其全部内容通过引用结合在本申请中。This application requires the priority of a Chinese patent submitted to the China Patent Office on May 5, 2022, with the application number 202210481719.6 and the application name "Rotor system dynamic characteristics calculation method, equipment and medium based on finite element method". All its contents have been approved. This reference is incorporated into this application.
技术领域Technical field
本申请涉及转子系统动力学统技术领域，尤其涉及基于有限元法的转子系统动特性计算方法、设备及介质。This application relates to the technical field of rotor system dynamics systems, and in particular to methods, equipment and media for calculating dynamic characteristics of rotor systems based on the finite element method.
背景技术Background technique
转子系统动力学的研究对象是横向位移远小于轴径(0.1％量级)的转子，其振动包括转轴扭转振动和弯曲振动、圆盘振动或盘片抖动等多种形式，其中转轴弯曲振动是最复杂，也是涉及因素最多的一种，因此转子系统动力学正是以转轴横向弯曲振动为主要研究对象。The research object of rotor system dynamics is a rotor whose lateral displacement is much smaller than the shaft diameter (on the order of 0.1%). Its vibration includes various forms such as torsional vibration and bending vibration of the rotating shaft, disc vibration or disc jitter, among which the bending vibration of the rotating shaft is It is the most complex and involves the most factors. Therefore, the rotor system dynamics takes the transverse bending vibration of the rotating shaft as the main research object.
随着旋转机械朝高转速、大功率及轻质量的方向发展，转子系统的非线性振动现象异常突出，复杂转子系统(含多自由度和强非线性)已成为现代转子动力学的主要研究对象。转子系统中同时存在多种非线性因素，如滚动轴承动刚度、密封阻尼和刚度、不平衡质量等，这些非线性因素会产生一定程度的耦合，导致自激振动、多解现象、拟周期运动和混沌运动的发生。这些运动状态是不平衡激励引起的强迫振动与非线性因素引发的低频运动的合成。With the development of rotating machinery towards high speed, high power and light weight, the nonlinear vibration phenomenon of the rotor system is extremely prominent. Complex rotor systems (including multiple degrees of freedom and strong nonlinearity) have become the main research object of modern rotor dynamics. . There are many nonlinear factors in the rotor system at the same time, such as rolling bearing dynamic stiffness, seal damping and stiffness, unbalanced mass, etc. These nonlinear factors will produce a certain degree of coupling, leading to self-excited vibration, multiple solution phenomena, quasi-periodic motion and The occurrence of chaotic motion. These motion states are a combination of forced vibration caused by unbalanced excitation and low-frequency motion caused by nonlinear factors.
对于复杂的非线性转子系统，完全采用解析方法求解十分困难。随着计算方法的改进和发展，数值计算方法已经广泛应用于非线性振动系统的研究，也是解决高维非线性动力学方程的最有效的方法。现代转子系统动力学的计算方 法可分为传递矩阵法和有限元法：传递矩阵法的特点是矩阵阶数不随系统自由度数的增大而增加，因此编程简单、内存量小、运算速度快，特别适用于转子这样的链式系统，但缺点是模拟轴承支承、密封等结构时非常困难；有限元法的特点是表达式规范，特别适用于转轴、轴承和密封等组成的复杂结构的分析。For complex nonlinear rotor systems, it is very difficult to solve them completely using analytical methods. With the improvement and development of calculation methods, numerical calculation methods have been widely used in the study of nonlinear vibration systems and are also the most effective methods for solving high-dimensional nonlinear dynamic equations. The calculation methods of modern rotor system dynamics can be divided into transfer matrix method and finite element method: The transfer matrix method is characterized by the fact that the matrix order does not increase with the increase of the system degrees of freedom, so it has simple programming, small memory and fast operation speed. It is especially suitable for chain systems such as rotors, but the disadvantage is that it is very difficult to simulate bearing supports, seals and other structures; the finite element method is characterized by expression specifications, and is especially suitable for the analysis of complex structures composed of rotating shafts, bearings and seals.
发明内容Contents of the invention
有鉴于此，本申请公开了一种基于有限元法的转子系统动特性计算方法、设备及介质，用于提供一种利用有限元法求解转子特性的过程和数值算法。In view of this, this application discloses a method, equipment and medium for calculating the dynamic characteristics of a rotor system based on the finite element method, which is used to provide a process and numerical algorithm for solving the rotor characteristics using the finite element method.
第一方面，本申请提供了一种基于有限元法的转子系统动特性计算方法，其特征在于，所述转子系统包括圆盘、轴承以及连接所述圆盘和轴承的弹性轴；所述基于有限元法的转子系统动特性计算方法包括：In the first aspect, this application provides a method for calculating the dynamic characteristics of a rotor system based on the finite element method, which is characterized in that the rotor system includes a disc, a bearing, and an elastic shaft connecting the disc and the bearing; the method is based on the finite element method. The calculation methods of the dynamic characteristics of the rotor system using the finite element method include:
基于欧拉角转动变换矩阵，计算所述圆盘的运动方程；Based on the Euler angle rotation transformation matrix, calculate the motion equation of the disk;
利用位移插值函数矩阵，计算所述弹性轴的运动方程；Use the displacement interpolation function matrix to calculate the motion equation of the elastic axis;
基于轴承的轴颈中心坐标，计算所述轴承的运动方程；Calculate the motion equation of the bearing based on the journal center coordinates of the bearing;
基于所述圆盘的运动方程、所述弹性轴的运动方程以及所述轴承的运动方程，确定所述转子系统的运动方程；Determine an equation of motion of the rotor system based on the equation of motion of the disk, the equation of motion of the elastic shaft, and the equation of motion of the bearing;
基于所述转子系统的运动方程，计算所述转子系统的临界转速和不平衡响应。Based on the equation of motion of the rotor system, the critical speed and unbalance response of the rotor system are calculated.
在采用上述技术方案的情况下，本申请首先基于欧拉角转动变换矩阵，计算圆盘的运动方程，基于位移插值函数矩阵，计算弹性轴的运动方程，基于轴承的轴颈中心坐标，计算轴承的运动方程，然后，基于所述圆盘的运动方程、弹性轴的运动方程以及轴承的运动方程，确定转子系统的运动方程，最后，基于转子系统的运动方程，计算转子系统的临界转速和不平衡响应。基于此，本申请能够提供的基于有限元法的转子系统动特性计算方法能够利用有限元法求解转子系统的转动过程中的运动方程以及计算出转子系统的临界转速和不平衡响应。In the case of adopting the above technical solution, this application first calculates the motion equation of the disk based on the Euler angle rotation transformation matrix, calculates the motion equation of the elastic shaft based on the displacement interpolation function matrix, and calculates the bearing journal center coordinates based on the The equation of motion of the rotor system is then determined based on the equation of motion of the disk, the equation of motion of the elastic shaft and the equation of motion of the bearing. Finally, based on the equation of motion of the rotor system, the critical rotational speed and inconsistency of the rotor system are calculated. Balanced response. Based on this, the method for calculating the dynamic characteristics of the rotor system based on the finite element method provided by this application can use the finite element method to solve the equations of motion during the rotation of the rotor system and calculate the critical speed and unbalance response of the rotor system.
应理解，转子系统中同时存在多种非线性因素，如滚动轴承动刚度、密封 阻尼和刚度、不平衡质量等，这些非线性因素会产生一定程度的耦合，导致自激振动、多解现象、拟周期运动和混沌运动的发生。这些运动状态是不平衡激励引起的强迫振动与非线性因素引发的低频运动的合成。对于复杂的非线性转子系统，完全采用解析方法求解十分困难。而本申请基于有限元法，能够把连续的转子系统划分成有限个单元，把单元的节点作为离散点，不考虑微分方程，特别适用于转轴、轴承和密封等组成的复杂结构的分析。It should be understood that there are many nonlinear factors in the rotor system at the same time, such as rolling bearing dynamic stiffness, seal damping and stiffness, unbalanced mass, etc. These nonlinear factors will produce a certain degree of coupling, leading to self-excited vibration, multiple solution phenomena, and pseudo The occurrence of periodic motion and chaotic motion. These motion states are a combination of forced vibration caused by unbalanced excitation and low-frequency motion caused by nonlinear factors. For complex nonlinear rotor systems, it is very difficult to solve them completely using analytical methods. This application is based on the finite element method, which can divide the continuous rotor system into a finite number of units, using the nodes of the units as discrete points without considering differential equations. It is especially suitable for the analysis of complex structures composed of rotating shafts, bearings, seals, etc.
第二方面，本申请实施例提供了一种基于有限元法的转子系统动特性计算设备，包括处理器以及与处理器耦合的通信接口；所述处理器用于运行计算机程序或指令，以实现基于有限元法的转子系统动特性计算方法。In the second aspect, embodiments of the present application provide a device for calculating dynamic characteristics of a rotor system based on the finite element method, including a processor and a communication interface coupled to the processor; the processor is used to run computer programs or instructions to implement calculations based on Finite element method is used to calculate the dynamic characteristics of the rotor system.
第三方面，本申请实施例提供了一种计算机存储介质，所述计算机存储介质中存储有指令，当所述指令被运行时，实现基于有限元法的转子系统动特性计算方法。In a third aspect, embodiments of the present application provide a computer storage medium. Instructions are stored in the computer storage medium. When the instructions are executed, a method for calculating the dynamic characteristics of the rotor system based on the finite element method is implemented.
与现有技术相比，本申请第二方面以及第三方面与上述技术方案考核评价方法的有益效果相同，此处不做赘述。Compared with the prior art, the second aspect and the third aspect of the present application have the same beneficial effects as the above-mentioned technical solution assessment and evaluation method, and will not be described again here.
附图说明Description of the drawings
为了更清楚地说明本申请实施例或现有技术中的技术方案，下面将对实施例或现有技术描述中所需要使用的附图进行说明：In order to more clearly explain the embodiments of the present application or the technical solutions in the prior art, the following will describe the drawings needed to describe the embodiments or the prior art:
图1为本申请实施例提供的一种基于有限元法的转子系统动特性计算方法的步骤流程图；Figure 1 is a step flow chart of a method for calculating dynamic characteristics of a rotor system based on the finite element method provided by an embodiment of the present application;
图2为本申请实施例提供的一种滚动轴承支承的转子系统示意图；Figure 2 is a schematic diagram of a rotor system supported by a rolling bearing provided in an embodiment of the present application;
图3为本申请实施例提供的一种轴段单元的示意图；Figure 3 is a schematic diagram of a shaft segment unit provided by an embodiment of the present application;
图4为本申请实施例提供的一种基于有限元法的转子系统动特性计算设备的硬件结构示意图；Figure 4 is a schematic diagram of the hardware structure of a rotor system dynamic characteristics calculation device based on the finite element method provided by an embodiment of the present application;
图5为本申请实施例提供的一种芯片的结构示意图。FIG. 5 is a schematic structural diagram of a chip provided by an embodiment of the present application.
附图标记：801-1-处理器，801-2-处理器，801-处理器，802-通信接口，803-通信线路，804-存储器，805-总线系统，90-芯片。Reference signs: 801-1-processor, 801-2-processor, 801-processor, 802-communication interface, 803-communication line, 804-memory, 805-bus system, 90-chip.
具体实施方式Detailed ways
本申请的核心是提供了一种基于有限元法的转子系统动特性计算方法。图1示出了本申请实施例提供的一种基于有限元法的转子系统动特性计算方法的步骤流程图。The core of this application is to provide a calculation method for the dynamic characteristics of the rotor system based on the finite element method. Figure 1 shows a step flow chart of a method for calculating dynamic characteristics of a rotor system based on the finite element method provided by an embodiment of the present application.
其中，参照图2，转子系统包括圆盘、轴承以及连接圆盘和轴承的弹性轴，其中图2中的轴承为滚动轴承。图中，l为弹性轴的长度，O 1、O 3为滚动轴承支撑处的轴颈中心，O 2为圆盘处的转子中心，圆盘的转轴任意截面处的位置可由轴心坐标x、y，截面转角θ x、θ y以及自转角
Figure PCTCN2022108728-appb-000001
来计算。 Referring to Figure 2, the rotor system includes a disc, a bearing and an elastic shaft connecting the disc and the bearing, where the bearing in Figure 2 is a rolling bearing. In the figure, l is the length of the elastic shaft, O 1 and O 3 are the journal centers at the rolling bearing support, O 2 is the rotor center at the disk. The position of the disk's rotating shaft at any cross-section can be determined by the axis coordinates x, y , section rotation angle θ x , θ y and rotation angle
Figure PCTCN2022108728-appb-000001
to calculate.
参照图1，上述基于有限元法的转子系统动特性计算方法包括以下步骤：Referring to Figure 1, the above-mentioned calculation method of dynamic characteristics of the rotor system based on the finite element method includes the following steps:
S100，基于欧拉角转动变换矩阵计算所述圆盘的运动方程。S100. Calculate the motion equation of the disk based on the Euler angular rotation transformation matrix.
此步骤具体包括：S101，当所述圆盘的轴心与所述圆盘的重心重合时，计算所述圆盘的动能；设圆盘轴心与重心重合，其轴心的位移向量为{u 1d}＝[x，θ y] T和{u 2d}＝[y，-θ x] T，则其动能为 This step specifically includes: S101, when the axis of the disk coincides with the center of gravity of the disk, calculate the kinetic energy of the disk; assuming that the axis of the disk coincides with the center of gravity, the displacement vector of its axis is { u 1d } = [x, θ y ] T and {u 2d } = [y, -θ x ] T , then its kinetic energy is
Figure PCTCN2022108728-appb-000002
Figure PCTCN2022108728-appb-000002
式中，x、y分别表示所述圆盘的轴心横坐标和纵坐标，
Figure PCTCN2022108728-appb-000003
分别表示所述圆盘的轴心横坐标和纵坐标对时间的一阶倒数，m d、J d和J p分别为刚性圆盘的质量、直径转动惯量和极转动惯量，O 2ξηζ是以轴心节点为原点，O 2ζ轴与圆盘平面垂直，固结于圆盘的动坐标系，ω ξ，ω η，ω ζ分别表示转速在所述动坐标系的第一分量，第二分量和第三分量。 In the formula, x and y respectively represent the abscissa and ordinate of the axis of the disk,
Figure PCTCN2022108728-appb-000003
represent the first-order reciprocal of the abscissa and ordinate of the axis of the disk to time respectively, m d , J d and J p are the mass, diameter moment of inertia and polar moment of inertia of the rigid disc respectively, O 2 ξηζ is The axis node is the origin, and the O 2 ζ axis is perpendicular to the disk plane and is fixed in the moving coordinate system of the disk. ω ξ , ω η , and ω ζ respectively represent the first component and the second component of the rotational speed in the moving coordinate system. component and third component.
S102，基于欧拉角转动变换矩阵以及所述圆盘的动能，计算所述圆盘的运 动方程。S102. Calculate the motion equation of the disk based on the Euler angular rotation transformation matrix and the kinetic energy of the disk.
根据欧拉角转动变换矩阵，可得According to the Euler angle rotation transformation matrix, we can get
Figure PCTCN2022108728-appb-000004
Figure PCTCN2022108728-appb-000004
式中，
Figure PCTCN2022108728-appb-000005
和
Figure PCTCN2022108728-appb-000006
分别表示圆盘中心绕O 2x、O 2y、O 2ξ以及O 2ζ轴转动角速度，
Figure PCTCN2022108728-appb-000007
等于圆盘转动角速度Ω。 In the formula,
Figure PCTCN2022108728-appb-000005
and
Figure PCTCN2022108728-appb-000006
represent the rotation angular velocity of the disk center around the O 2 x, O 2 y, O 2 ξ and O 2 ζ axes respectively,
Figure PCTCN2022108728-appb-000007
It is equal to the angular velocity of disk rotation Ω.
将式(6-2)代入式(6-1)，略去二阶及以上微量，可得圆盘的运动方程为：Substituting Equation (6-2) into Equation (6-1), omitting the second-order and above traces, the motion equation of the disk can be obtained as:
Figure PCTCN2022108728-appb-000008
Figure PCTCN2022108728-appb-000008
式中，
Figure PCTCN2022108728-appb-000009
Figure PCTCN2022108728-appb-000010
分别表示圆盘轴心位移向量的第一分量和第二分量对时间的一阶导数，M d表示圆盘的广义质量，Ω表示圆盘转动角速度，J表示圆盘的转动惯量。 In the formula,
Figure PCTCN2022108728-appb-000009
Figure PCTCN2022108728-appb-000010
represent the first derivative of the first component and the second component of the disk axis displacement vector with respect to time respectively, M d represents the generalized mass of the disk, Ω represents the rotation angular velocity of the disk, and J represents the rotational inertia of the disk.
对于具有n个自由度的动力系统，其状态和位置的Lagrange方程为For a dynamic system with n degrees of freedom, the Lagrange equation of its state and position is
Figure PCTCN2022108728-appb-000011
Figure PCTCN2022108728-appb-000011
式中，T为系统的动能，q i、
Figure PCTCN2022108728-appb-000012
分别为广义位移和广义速度。 In the formula, T is the kinetic energy of the system, q i ,
Figure PCTCN2022108728-appb-000012
are generalized displacement and generalized velocity respectively.
根据Lagrange方程式(6-4)，将式圆盘的运动方程改写为：According to Lagrange's equation (6-4), the motion equation of the disc is rewritten as:
Figure PCTCN2022108728-appb-000013
Figure PCTCN2022108728-appb-000013
式中，
Figure PCTCN2022108728-appb-000014
分别表示圆盘轴心位移向量的第一分量和第二分量对时间的二阶导数，{Q 1d}＝[F x，M y] T，{Q 2d}＝[F y，M x] T；其中，Q 1d，Q 2d分别表示法向接 触载荷，F x，F y分别表示所述圆盘受到的外力的第一分量和第二分量，M x，M y分别表示力矩的第一分量和第二分量。 In the formula,
Figure PCTCN2022108728-appb-000014
Respectively represent the second derivative of the first component and the second component of the disk axis displacement vector with respect to time, {Q 1d } = [F x , M y ] T , {Q 2d } = [F y , M x ] T ; Among them, Q 1d and Q 2d respectively represent the normal contact load, F x and F y respectively represent the first and second components of the external force on the disk, and M x and M y respectively represent the first component of the moment. and the second component.
S200，利用位移插值函数矩阵，计算所述弹性轴的运动方程。S200, use the displacement interpolation function matrix to calculate the motion equation of the elastic axis.
此步骤具体包括：S201，将所述弹性轴分割为多个轴段单元。This step specifically includes: S201, dividing the elastic shaft into multiple shaft segment units.
图3示出了轴段单元的结构示意图，参照图3，轴段单元的广义坐标是两节点的位移，即Figure 3 shows a schematic structural diagram of the shaft segment unit. Referring to Figure 3, the generalized coordinates of the shaft segment unit are the displacements of the two nodes, that is,
Figure PCTCN2022108728-appb-000015
Figure PCTCN2022108728-appb-000015
其中，x A，x B，y A，y B，θ xA，θ xB，θ yA，θ yB分别表示A点和B点沿X方向的坐标，Y方向的坐标，X方向的夹角，Y方向的夹角。 Among them, x A , x B , y A , y B , θ xA , θ xB , θ yA , θ yB respectively represent the coordinates of point A and point B along the X direction, the coordinates of the Y direction, the angle between the X direction, Y direction angle.
S202，利用位移插值函数矩阵，以及所述轴段单元的节点位移，计算所述轴段单元的动能和弯曲势能。S202: Calculate the kinetic energy and bending potential energy of the shaft segment unit using the displacement interpolation function matrix and the node displacement of the shaft segment unit.
由于轴段单元任一截面处的位移x、θ y、y和θ x是位置z和时间t的函数，故轴段单元任一截面处的位移可通过位移插值函数和轴段单元节点的位移来表示 Since the displacements x, θ y , y and θ x at any section of the shaft segment unit are functions of position z and time t, the displacement at any section of the shaft segment unit can be calculated through the displacement interpolation function and the displacement of the node of the shaft segment unit. To represent
Figure PCTCN2022108728-appb-000016
Figure PCTCN2022108728-appb-000016
其中，[N]＝[N 1(z) N 2(z) N 3(z) N 4(z)]为1×4阶的位移插值函数矩阵。 Among them, [N]=[N 1 (z) N 2 (z) N 3 (z) N 4 (z)] is a 1×4 order displacement interpolation function matrix.
根据轴段单元的端点条件According to the end point conditions of the shaft segment element
x(0，t)＝x A(t)，x(l，t)＝x B(t)，x′(0，t)＝θ yA(t)，x′(l，t)＝θ yB(t)     (6-8) x (0, t) = x A (t), x (l, t) = x B (t), x' (0, t) = θ yA (t), x' (l, t) = θ yB (t) (6-8)
可知插值函数满足It can be seen that the interpolation function satisfies
Figure PCTCN2022108728-appb-000017
Figure PCTCN2022108728-appb-000017
可解得位移插值函数为The displacement interpolation function can be solved as
Figure PCTCN2022108728-appb-000018
Figure PCTCN2022108728-appb-000018
综合以上结果，可得轴段单元任一截面处的位移的表达式为：Based on the above results, the expression of the displacement at any section of the shaft segment unit can be obtained as:
Figure PCTCN2022108728-appb-000019
Figure PCTCN2022108728-appb-000019
其中，x(z，t)，y(z，t)分别表示所述轴段单元任一截面处的位移、θ x(z，t)，θ y(z，t)表示所述轴段单元任一截面处的角位移，u 1z，u 2z分别表示直角坐标系下圆盘转度的第三分量，z表示位置，t表示时间，N为位移插值函数； Among them, x (z, t), y (z, t) respectively represent the displacement at any section of the shaft segment unit, θ x (z, t), θ y (z, t) represent the shaft segment unit The angular displacement at any section, u 1z and u 2z respectively represent the third component of the disk rotation in the Cartesian coordinate system, z represents the position, t represents the time, and N is the displacement interpolation function;
因此轴段单元的任意点的位移可以用该单元节点的位移来表示，单元的动能和势能也可以表示为节点位移和速度的函数。根据式(6-3)，可得轴段微元的动能为：Therefore, the displacement of any point of the axis segment unit can be expressed by the displacement of the node of the unit, and the kinetic energy and potential energy of the unit can also be expressed as functions of the node displacement and velocity. According to formula (6-3), the kinetic energy of the shaft segment micro-element can be obtained:
Figure PCTCN2022108728-appb-000020
Figure PCTCN2022108728-appb-000020
式中，dm、j d、j p分别表示轴段微元的质量、直径转动惯量和极转动惯量。 In the formula, dm, j d and j p respectively represent the mass, diameter moment of inertia and polar moment of inertia of the shaft segment element.
将式(6-11)对时间求导数，并代入式(6-12)，可得Calculate the derivative of equation (6-11) with respect to time and substitute it into equation (6-12), we can get
Figure PCTCN2022108728-appb-000021
Figure PCTCN2022108728-appb-000021
其中，dm、j d、j p分别表示所述轴段单元的质量、直径转动惯量和极转动惯量，N’表示轴段单元的2次位移插值多项式。 Among them, dm, j d and j p respectively represent the mass, diameter moment of inertia and polar moment of inertia of the shaft segment unit, and N' represents the quadratic displacement interpolation polynomial of the shaft segment unit.
轴段微元的弯曲势能为The bending potential energy of the shaft segment elements
Figure PCTCN2022108728-appb-000022
Figure PCTCN2022108728-appb-000022
式中，E为弹性模量，I为横截面对弯曲中性轴的惯性矩，M”表示轴段单元的3次位移插值多项式，
Figure PCTCN2022108728-appb-000023
和
Figure PCTCN2022108728-appb-000024
分别表示轴段单元任一截面处的沿x、y方向的加速度。 In the formula, E is the elastic modulus, I is the moment of inertia of the cross section against the bending neutral axis, M” represents the third-order displacement interpolation polynomial of the axis segment unit,
Figure PCTCN2022108728-appb-000023
and
Figure PCTCN2022108728-appb-000024
Respectively represent the acceleration along the x and y directions at any section of the shaft segment unit.
对于长为l，半径为R的圆形截面转轴，将式(6-13)、式(6-14)沿全长积分，可以得到For a circular cross-section axis with length l and radius R, integrating equations (6-13) and (6-14) along the entire length, we can get
Figure PCTCN2022108728-appb-000025
Figure PCTCN2022108728-appb-000025
其中，M zT，M zR分别表示表示轴段单元的对角质量矩阵，J z表示轴段单元的转动惯量； Among them, M zT and M zR respectively represent the diagonal mass matrix representing the axis segment unit, and J z represents the moment of inertia of the axis segment unit;
Figure PCTCN2022108728-appb-000026
Figure PCTCN2022108728-appb-000026
K z表示轴段单元的刚度矩阵。 K z represents the stiffness matrix of the axis segment element.
Figure PCTCN2022108728-appb-000027
Figure PCTCN2022108728-appb-000027
Figure PCTCN2022108728-appb-000028
Figure PCTCN2022108728-appb-000028
Figure PCTCN2022108728-appb-000029
Figure PCTCN2022108728-appb-000029
Figure PCTCN2022108728-appb-000030
Figure PCTCN2022108728-appb-000030
S203，将所述轴段单元的动能和弯曲势能代入Lagrange方程中，得到所述轴段单元的运动方程。S203. Substitute the kinetic energy and bending potential energy of the shaft segment unit into the Lagrange equation to obtain the motion equation of the shaft segment unit.
将所述轴段单元的动能和弯曲势能代入Lagrange方程中，得到所述轴段 单元的运动方程包括：Substituting the kinetic energy and bending potential energy of the shaft segment unit into the Lagrange equation, the motion equation of the shaft segment unit is obtained:
Figure PCTCN2022108728-appb-000031
Figure PCTCN2022108728-appb-000031
式中，[M z]＝[M zR]+[M zT]为对角矩阵，{Q 1z}、{Q 2z}为对应的广义力向量，包括节点处连接的圆盘或相邻轴段的作用力、力矩以及不平衡力。 In the formula, [M z ]=[M zR ]+[M zT ] is a diagonal matrix, {Q 1z }, {Q 2z } are the corresponding generalized force vectors, including the disks or adjacent shaft segments connected at the nodes. forces, moments and unbalanced forces.
在已知偏心规律的情况下，可以得到轴段微元的不平衡力为：When the eccentricity law is known, the unbalanced force of the shaft segment element can be obtained as:
Figure PCTCN2022108728-appb-000032
Figure PCTCN2022108728-appb-000032
式中，e ξ(z)、e η(z)为轴段单元质量分布偏心距。 In the formula, e ξ (z) and e η (z) are the eccentricity of the mass distribution of the shaft segment unit.
S300，基于轴承的轴颈中心坐标，计算所述轴承的运动方程。S300: Calculate the motion equation of the bearing based on the journal center coordinates of the bearing.
对于轴承支承，其运动方程为For bearing support, its equation of motion is
Figure PCTCN2022108728-appb-000033
Figure PCTCN2022108728-appb-000033
式中，M bx，M by分别表示表示轴承在x、y方向上的质量，x b、y b为所述轴承的轴承座中心的坐标，
Figure PCTCN2022108728-appb-000034
为所述轴承的轴承座中心的坐标对时间的一阶倒数，
Figure PCTCN2022108728-appb-000035
为所述轴承的轴承座中心的坐标对时间的二阶倒数，x s(j)、y s(j)为所述轴承的轴颈中心的坐标，k bxx，k bxy，k byx，k byy分别表示轴承在不同方向上的刚度，c bxx，c bxy，c byx，c byy分别表示轴承在不同方向上的阻尼。 In the formula, M bx and M by respectively represent the mass of the bearing in the x and y directions, x b and y b are the coordinates of the bearing seat center of the bearing,
Figure PCTCN2022108728-appb-000034
is the first-order reciprocal of the coordinates of the bearing seat center of the bearing versus time,
Figure PCTCN2022108728-appb-000035
is the second-order reciprocal of the coordinates of the bearing seat center of the bearing versus time, x s(j) and y s(j) are the coordinates of the journal center of the bearing, k bxx , k bxy , k byx , k byy Respectively represent the stiffness of the bearing in different directions, c bxx , c bxy , c byx , c byy respectively represent the damping of the bearing in different directions.
S400，基于所述圆盘的运动方程、所述弹性轴的运动方程以及所述轴承的 运动方程，确定所述转子系统的运动方程。S400, determine the motion equation of the rotor system based on the motion equation of the disk, the motion equation of the elastic shaft, and the motion equation of the bearing.
对于具有n个节点，n-1个轴段单元的转子系统，综合圆盘、轴段单元以及轴承的运动方程，可得转子系统的运动方程为For a rotor system with n nodes and n-1 shaft segment units, integrating the motion equations of the disk, shaft segment units and bearings, the motion equation of the rotor system can be obtained as
Figure PCTCN2022108728-appb-000036
Figure PCTCN2022108728-appb-000036
式中，{U 1}＝[x 1，θ y1，x 2，θ y2，...，x n，θ yn] T、{U 2}＝[y 1，-θ x1，y 2，-θ x2，...，y n，-θ xn] T为转子系统的位移向量，
Figure PCTCN2022108728-appb-000037
为转子系统的位移向量对时间的一阶倒数，
Figure PCTCN2022108728-appb-000038
为转子系统的位移向量对时间的二阶倒数，[M]、Ω[J]、[K]分别为转子系统的质量矩阵、回转矩阵以及刚度矩阵，都是半带宽为4的2n×2n阶对称稀疏矩阵，{Q 1}、{Q 2}为转子系统的广义力。 In the formula, {U 1 } = [x 1 , θ y1 , x 2 , θ y2 ,..., x n , θ yn ] T , {U 2 } = [y 1 , -θ x1 , y 2 , - θ x2 ,...,y n ,-θ xn ] T is the displacement vector of the rotor system,
Figure PCTCN2022108728-appb-000037
is the first-order reciprocal of the displacement vector of the rotor system versus time,
Figure PCTCN2022108728-appb-000038
is the second-order reciprocal of the displacement vector of the rotor system versus time, [M], Ω[J], and [K] are the mass matrix, rotation matrix, and stiffness matrix of the rotor system respectively, all of order 2n×2n with a half-bandwidth of 4 Symmetric sparse matrices, {Q 1 }, {Q 2 } are the generalized forces of the rotor system.
式(6-25)可以写成统一形式为Equation (6-25) can be written in the unified form as
Figure PCTCN2022108728-appb-000039
Figure PCTCN2022108728-appb-000039
S500，基于所述转子系统的运动方程，计算所述转子系统的临界转速和不平衡响应。S500: Calculate the critical speed and unbalance response of the rotor system based on the motion equation of the rotor system.
转子系统的临界转速是指转子系统在自身的不平衡激振力作用下产生共振时的转速，临界转速特性是转子系统的固有特性。The critical speed of the rotor system refers to the speed when the rotor system resonates under its own unbalanced excitation force. The critical speed characteristics are the inherent characteristics of the rotor system.
用有限元法建立了转子系统的运动方程后，可通过微分方程的齐次解，求出自转角速度为Ω时的涡轮频率，也可以求得当Ω＝ω时转子系统的临界转速及相应振型。After establishing the motion equation of the rotor system using the finite element method, the turbine frequency when the rotational angular velocity is Ω can be obtained through the homogeneous solution of the differential equation. The critical speed and corresponding vibration shape of the rotor system when Ω=ω can also be obtained. .
当支承为滚动轴承时，即k xx≠k yy，k xy＝k yx＝0，不计阻尼影响，转子系统的运动方程齐次式为 When the support is a rolling bearing, that is, k xx ≠k yy , k xy =k yx =0, regardless of the damping effect, the homogeneous equation of motion of the rotor system is:
Figure PCTCN2022108728-appb-000040
Figure PCTCN2022108728-appb-000040
式中，
Figure PCTCN2022108728-appb-000041
[M]、[K}为实对称矩阵，[J]为实反对称矩阵。 In the formula,
Figure PCTCN2022108728-appb-000041
[M] and [K} are real symmetric matrices, and [J] is a real antisymmetric matrix.
则频率方程可得Then the frequency equation can be obtained
|-[M]ω 2+Ω[J]ω+[K]|＝0          (6-28) |-[M]ω 2 +Ω[J]ω+[K]|＝0 (6-28)
该方程的特征值是由4n对共轭的复数组成，分别对应于2n个正向涡动频率和2n个反向涡动频率。The eigenvalues of this equation are composed of 4n pairs of conjugate complex numbers, corresponding to 2n forward whirl frequencies and 2n reverse whirl frequencies respectively.
(2)不平衡响应的分析(2) Analysis of unbalanced response
转子系统在不平衡力或不平衡力矩的激励下所产生的振动称为不平衡响应，主要用于研究转子系统对某些位置上不平衡量的敏感程度。The vibration generated by the rotor system under the excitation of unbalanced force or unbalanced torque is called unbalanced response, which is mainly used to study the sensitivity of the rotor system to the unbalanced amount at certain positions.
对于各向同性轴承，不计阻尼和轴承座振动的影响，转子系统的不平衡响应可由式(6-25)得到For isotropic bearings, regardless of the effects of damping and bearing seat vibration, the unbalanced response of the rotor system can be obtained from Equation (6-25)
Figure PCTCN2022108728-appb-000042
Figure PCTCN2022108728-appb-000042
式中，{z}＝{U 1}+{U 2}，{Q}＝{Q 1c}+i{Q 2c}表示不平衡力，i表示虚数，Q 1c和Q 2c分别表示不平衡力在复数平面上分解值。 In the formula, {z}={U 1 }+{U 2 }, {Q}={Q 1c }+i{Q 2c } represents the unbalanced force, i represents the imaginary number, Q 1c and Q 2c represent the unbalanced force respectively. Decompose values on the complex plane.
基于以上描述，本申请实施例首先基于欧拉角转动变换矩阵，计算圆盘的运动方程，基于位移插值函数矩阵，计算弹性轴的运动方程，基于轴承的轴颈中心坐标，计算轴承的运动方程，然后，基于所述圆盘的运动方程、弹性轴的运动方程以及轴承的运动方程，确定转子系统的运动方程，最后，基于转子系统的运动方程，计算转子系统的临界转速和不平衡响应。基于此，本申请实施例能够提供的基于有限元法的转子系统动特性计算方法能够利用有限元法求解转子系统的转动过程中的运动方程以及计算出转子系统的临界转速和不平 衡响应。Based on the above description, the embodiment of the present application first calculates the motion equation of the disk based on the Euler angle rotation transformation matrix, calculates the motion equation of the elastic shaft based on the displacement interpolation function matrix, and calculates the motion equation of the bearing based on the journal center coordinates of the bearing. , then, determine the motion equation of the rotor system based on the motion equation of the disk, the elastic shaft, and the bearing. Finally, based on the motion equation of the rotor system, calculate the critical speed and unbalance response of the rotor system. Based on this, the method for calculating the dynamic characteristics of the rotor system based on the finite element method provided by the embodiments of the present application can use the finite element method to solve the equations of motion during the rotation of the rotor system and calculate the critical speed and unbalanced response of the rotor system.
应理解，转子系统中同时存在多种非线性因素，如滚动轴承动刚度、密封阻尼和刚度、不平衡质量等，这些非线性因素会产生一定程度的耦合，导致自激振动、多解现象、拟周期运动和混沌运动的发生。这些运动状态是不平衡激励引起的强迫振动与非线性因素引发的低频运动的合成。对于复杂的非线性转子系统，完全采用解析方法求解十分困难。而本申请实施例基于有限元法，能够把连续的转子系统划分成有限个单元，把单元的节点作为离散点，不考虑微分方程，特别适用于转轴、轴承和密封等组成的复杂结构的分析。It should be understood that there are many nonlinear factors in the rotor system at the same time, such as rolling bearing dynamic stiffness, seal damping and stiffness, unbalanced mass, etc. These nonlinear factors will produce a certain degree of coupling, leading to self-excited vibration, multiple solution phenomena, and pseudo The occurrence of periodic motion and chaotic motion. These motion states are a combination of forced vibration caused by unbalanced excitation and low-frequency motion caused by nonlinear factors. For complex nonlinear rotor systems, it is very difficult to solve them completely using analytical methods. The embodiment of this application is based on the finite element method, which can divide the continuous rotor system into a finite number of units, and use the nodes of the units as discrete points without considering differential equations. It is especially suitable for the analysis of complex structures composed of rotating shafts, bearings, seals, etc. .
图4示出了本申请实施例提供的一种基于有限元法的转子系统动特性计算设备的硬件结构示意图。如图4所示，该基于基于有限元法的转子系统动特性计算设备80包括处理器801和通信接口802。Figure 4 shows a schematic diagram of the hardware structure of a device for calculating dynamic characteristics of a rotor system based on the finite element method provided by an embodiment of the present application. As shown in FIG. 4 , the rotor system dynamic characteristics calculation device 80 based on the finite element method includes a processor 801 and a communication interface 802 .
如图4所示，上述处理器可以是一个通用中央处理器(central processing unit，CPU)，微处理器，专用集成电路(application-specific integrated circuit，ASIC)，或一个或多个用于控制本申请方案程序执行的集成电路。上述通信接口可以为一个或多个。通信接口可使用任何收发器一类的装置，用于与其他设备或通信网络通信。As shown in Figure 4, the above-mentioned processor can be a general central processing unit (CPU), a microprocessor, an application-specific integrated circuit (ASIC), or one or more processors for controlling the computer. The application program program is implemented on an integrated circuit. There may be one or more communication interfaces. The communication interface may use any device such as a transceiver for communicating with other devices or communication networks.
如图4所示，上述基于有限元法的转子系统动特性计算设备还可以包括通信线路803。通信线路可包括一通路，在上述组件之间传送信息。As shown in FIG. 4 , the above-mentioned finite element method-based rotor system dynamic characteristics calculation device may also include a communication line 803 . The communication line may include a path to carry information between the above-mentioned components.
可选的，如图4所示，该基于有限元法的转子系统动特性计算设备还可以包括存储器804。存储器用于存储执行本申请方案的计算机执行指令，并由处理器来控制执行。处理器用于执行存储器中存储的计算机执行指令，从而实现本申请实施例提供的方法。Optionally, as shown in FIG. 4 , the finite element method-based rotor system dynamic characteristics calculation device may also include a memory 804 . The memory is used to store computer execution instructions for executing the solution of the present application, and is controlled by the processor for execution. The processor is used to execute computer execution instructions stored in the memory, thereby implementing the method provided by the embodiment of the present application.
如图4所示，上述存储器可以是只读存储器(read-only memory，ROM)或可存储静态信息和指令的其他类型的静态存储设备，随机存取存储器(random access memory，RAM)或者可存储信息和指令的其他类型的动态存储设备，也可以是电可擦可编程只读存储器(electrically erasable programmable read-only memory，EEPROM)、只读光盘(compact disc read-only memory，CD-ROM)或其他光盘存储、光碟存储(包括压缩光碟、激光碟、光碟、数字 通用光碟、蓝光光碟等)、磁盘存储介质或者其他磁存储设备、或者能够用于携带或存储具有指令或数据结构形式的期望的程序代码并能够由计算机存取的任何其他介质，但不限于此。存储器可以是独立存在，通过通信线路与处理器相连接。存储器也可以和处理器集成在一起。As shown in Figure 4, the above-mentioned memory can be a read-only memory (ROM) or other types of static storage devices that can store static information and instructions, a random access memory (random access memory, RAM) or a Other types of dynamic storage devices for information and instructions, which may also be electrically erasable programmable read-only memory (EEPROM), compact disc read-only memory (CD-ROM) or Other optical disc storage, optical disc storage (including compressed optical discs, laser discs, optical discs, digital versatile discs, Blu-ray discs, etc.), magnetic disk storage media or other magnetic storage devices, or can be used to carry or store desired information in the form of instructions or data structures Program code and any other medium capable of being accessed by a computer, without limitation. The memory can exist independently and be connected to the processor through communication lines. Memory can also be integrated with the processor.
可选的，本申请实施例中的计算机执行指令也可以称之为应用程序代码，本申请实施例对此不作具体限定。Optionally, the computer-executed instructions in the embodiments of the present application may also be called application codes, which are not specifically limited in the embodiments of the present application.
在具体实现中，作为一种实施例，如图4所示，处理器801可以包括一个或多个CPU，如图4中的CPU0和CPU1。In specific implementation, as an embodiment, as shown in FIG. 4 , the processor 801 may include one or more CPUs, such as CPU0 and CPU1 in FIG. 4 .
在具体实现中，作为一种实施例，如图4所示，基于有限元法的转子系统动特性计算设备可以包括多个处理器，如图4中的处理器801-1和处理器801-2。这些处理器中的每一个可以是一个单核处理器，也可以是一个多核处理器。In specific implementation, as an embodiment, as shown in Figure 4, the rotor system dynamic characteristics calculation device based on the finite element method may include multiple processors, such as processor 801-1 and processor 801- in Figure 4 2. Each of these processors can be a single-core processor or a multi-core processor.
图5是本申请实施例提供的芯片的结构示意图。如图5所示，该芯片90包括一个或两个以上(包括两个)处理器801和通信接口802。FIG. 5 is a schematic structural diagram of a chip provided by an embodiment of the present application. As shown in FIG. 5 , the chip 90 includes one or more (including two) processors 801 and a communication interface 802 .
可选的，如图5所示，该芯片还包括存储器804，存储器可以包括只读存储器和随机存取存储器，并向处理器提供操作指令和数据。存储器的一部分还可以包括非易失性随机存取存储器(non-volatile random access memory，NVRAM)。Optionally, as shown in Figure 5, the chip also includes a memory 804, which can include read-only memory and random access memory, and provides operating instructions and data to the processor. Part of the memory may also include non-volatile random access memory (NVRAM).
在一些实施方式中，如图5所示，存储器存储了如下的元素，执行模块或者数据结构，或者他们的子集，或者他们的扩展集。In some embodiments, as shown in Figure 5, the memory stores the following elements, execution modules or data structures, or their subsets, or their extended sets.
在本申请实施例中，如图5所示，通过调用存储器存储的操作指令(该操作指令可存储在操作系统中)，执行相应的操作。In the embodiment of the present application, as shown in FIG. 5 , the corresponding operation is performed by calling the operation instructions stored in the memory (the operation instructions can be stored in the operating system).
如图5所示，处理器控制基于有限元法的转子系统动特性计算设备中任一个的处理操作，处理器还可以称为中央处理单元(central processing unit，CPU)。As shown in Figure 5, the processor controls the processing operations of any one of the rotor system dynamic characteristics calculation devices based on the finite element method. The processor can also be called a central processing unit (CPU).
如图5所示，存储器可以包括只读存储器和随机存取存储器，并向处理器提供指令和数据。存储器的一部分还可以包括NVRAM。例如应用中存储器、通信接口以及存储器通过总线系统耦合在一起，其中总线系统除包括数据总线之外，还可以包括电源总线、控制总线和状态信号总线等。但是为了清楚说明 起见，在图5中将各种总线都标为总线系统805。As shown in Figure 5, memory may include read-only memory and random access memory and provide instructions and data to the processor. Part of the memory may also include NVRAM. For example, in an application, the memory, communication interface and memory are coupled together through a bus system. In addition to the data bus, the bus system may also include a power bus, a control bus, a status signal bus, etc. However, for the sake of clarity, the various buses are labeled bus system 805 in Figure 5.
如图5所示，上述本申请实施例揭示的方法可以应用于处理器中，或者由处理器实现。处理器可能是一种集成电路芯片，具有信号的处理能力。在实现过程中，上述方法的各步骤可以通过处理器中的硬件的集成逻辑电路或者软件形式的指令完成。上述的处理器可以是通用处理器、数字信号处理器(digital signal processing，DSP)、ASIC、现成可编程门阵列(field-programmable gate array，FPGA)或者其他可编程逻辑器件、分立门或者晶体管逻辑器件、分立硬件组件。可以实现或者执行本申请实施例中的公开的各方法、步骤及逻辑框图。通用处理器可以是微处理器或者该处理器也可以是任何常规的处理器等。结合本申请实施例所公开的方法的步骤可以直接体现为硬件译码处理器执行完成，或者用译码处理器中的硬件及软件模块组合执行完成。软件模块可以位于随机存储器，闪存、只读存储器，可编程只读存储器或者电可擦写可编程存储器、寄存器等本领域成熟的存储介质中。该存储介质位于存储器，处理器读取存储器中的信息，结合其硬件完成上述方法的步骤。As shown in FIG. 5 , the method disclosed in the above embodiment of the present application can be applied in a processor or implemented by the processor. The processor may be an integrated circuit chip that has signal processing capabilities. During the implementation process, each step of the above method can be completed by instructions in the form of hardware integrated logic circuits or software in the processor. The above-mentioned processor can be a general-purpose processor, digital signal processing (DSP), ASIC, off-the-shelf programmable gate array (field-programmable gate array, FPGA) or other programmable logic devices, discrete gates or transistor logic. devices, discrete hardware components. Each method, step and logical block diagram disclosed in the embodiment of this application can be implemented or executed. A general-purpose processor may be a microprocessor or the processor may be any conventional processor, etc. The steps of the method disclosed in conjunction with the embodiments of the present application can be directly implemented by a hardware decoding processor, or executed by a combination of hardware and software modules in the decoding processor. The software module can be located in random access memory, flash memory, read-only memory, programmable read-only memory or electrically erasable programmable memory, registers and other mature storage media in this field. The storage medium is located in the memory, and the processor reads the information in the memory and completes the steps of the above method in combination with its hardware.
一种可能的实现方式中，如图5所示，通信接口用于获得相机采集的图像。处理器用于执行图1所示的实施例中的考核评价方法的步骤101至步骤103。In one possible implementation, as shown in Figure 5, the communication interface is used to obtain images collected by the camera. The processor is used to execute steps 101 to 103 of the assessment and evaluation method in the embodiment shown in FIG. 1 .
一方面，提供一种计算机可读存储介质，计算机可读存储介质中存储有指令，当指令被运行时，实现上述实施例中由基于有限元法的转子系统动特性计算设备执行的功能。On the one hand, a computer-readable storage medium is provided. Instructions are stored in the computer-readable storage medium. When the instructions are executed, the functions performed by the rotor system dynamic characteristics calculation device based on the finite element method in the above embodiments are realized.
一方面，提供一种芯片，该芯片应用于基于有限元法的转子系统动特性计算设备中，芯片包括至少一个处理器和通信接口，通信接口和至少一个处理器耦合，处理器用于运行指令，以实现上述实施例中由基于有限元法的转子系统动特性计算设备执行的功能。On the one hand, a chip is provided. The chip is used in a rotor system dynamic characteristics calculation device based on the finite element method. The chip includes at least one processor and a communication interface. The communication interface is coupled with at least one processor. The processor is used to run instructions. In order to realize the functions performed by the rotor system dynamic characteristics calculation device based on the finite element method in the above embodiments.
在上述实施例中，可以全部或部分地通过软件、硬件、固件或者其任意组合来实现。当使用软件实现时，可以全部或部分地以计算机程序产品的形式实现。所述计算机程序产品包括一个或多个计算机程序或指令。在计算机上加载和执行所述计算机程序或指令时，全部或部分地执行本申请实施例所述的流程或功能。所述计算机可以是通用计算机、专用计算机、计算机网络、终端、用 户设备或者其它可编程装置。所述计算机程序或指令可以存储在计算机可读存储介质中，或者从一个计算机可读存储介质向另一个计算机可读存储介质传输，例如，所述计算机程序或指令可以从一个网站站点、计算机、服务器或数据中心通过有线或无线方式向另一个网站站点、计算机、服务器或数据中心进行传输。所述计算机可读存储介质可以是计算机能够存取的任何可用介质或者是集成一个或多个可用介质的服务器、数据中心等数据存储设备。所述可用介质可以是磁性介质，例如，软盘、硬盘、磁带；也可以是光介质，例如，数字视频光盘(digital video disc，DVD)；还可以是半导体介质，例如，固态硬盘(solid state drive，SSD)。In the above embodiments, it may be implemented in whole or in part by software, hardware, firmware, or any combination thereof. When implemented using software, it may be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer programs or instructions. When the computer program or instructions are loaded and executed on the computer, the processes or functions described in the embodiments of the present application are executed in whole or in part. The computer may be a general purpose computer, a special purpose computer, a computer network, a terminal, a user equipment, or other programmable device. The computer program or instructions may be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer program or instructions may be transmitted from a website, computer, A server or data center transmits via wired or wireless means to another website site, computer, server, or data center. The computer-readable storage medium may be any available medium that can be accessed by a computer or a data storage device such as a server or data center that integrates one or more available media. The available media may be magnetic media, such as floppy disks, hard disks, and magnetic tapes; they may also be optical media, such as digital video discs (DVDs); they may also be semiconductor media, such as solid state drives (solid state drives). ,SSD).
尽管在此结合各实施例对本申请进行了描述，然而，在实施所要求保护的本申请过程中，本领域技术人员通过查看附图、公开内容、以及所附权利要求书，可理解并实现公开实施例的其他变化。在权利要求中，“包括”(comprising)一词不排除其他组成部分或步骤，“一”或“一个”不排除多个的情况。单个处理器或其他单元可以实现权利要求中列举的若干项功能。相互不同的从属权利要求中记载了某些措施，但这并不表示这些措施不能组合起来产生良好的效果。Although the present application has been described herein in connection with various embodiments, in practicing the claimed application, those skilled in the art can understand and implement the disclosure by reviewing the drawings, the disclosure, and the appended claims. Other variations of the embodiment. In the claims, the word "comprising" does not exclude other components or steps, and "a" or "an" does not exclude a plurality. A single processor or other unit may perform several of the functions recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not mean that a combination of these measures cannot be combined to advantageous effects.
尽管结合具体特征及其实施例对本申请进行了描述，显而易见的，在不脱离本申请的精神和范围的情况下，可对其进行各种修改和组合。相应地，本说明书和附图仅仅是所附权利要求所界定的本申请的示例性说明，且视为已覆盖本申请范围内的任意和所有修改、变化、组合或等同物。显然，本领域的技术人员可以对本申请进行各种改动和变型而不脱离本申请的精神和范围。这样，倘若本申请的这些修改和变型属于本申请权利要求及其等同技术的范围之内，则本申请也意图包括这些改动和变型在内。Although the present application has been described in conjunction with specific features and embodiments thereof, it will be apparent that various modifications and combinations may be made without departing from the spirit and scope of the application. Accordingly, the specification and drawings are intended to be merely illustrative of the application as defined by the appended claims and are to be construed to cover any and all modifications, variations, combinations or equivalents within the scope of the application. Obviously, those skilled in the art can make various changes and modifications to the present application without departing from the spirit and scope of the present application. In this way, if these modifications and variations of the present application fall within the scope of the claims of the present application and its equivalent technology, the present application is also intended to include these modifications and variations.

Claims (10)

  1. 一种基于有限元法的转子系统动特性计算方法，其特征在于，所述转子系统包括圆盘、轴承以及连接所述圆盘和轴承的弹性轴；所述基于有限元法的转子系统动特性计算方法包括：A method for calculating the dynamic characteristics of a rotor system based on the finite element method, characterized in that the rotor system includes a disc, a bearing and an elastic shaft connecting the disc and the bearing; the dynamic characteristics of the rotor system based on the finite element method Calculation methods include:
    基于欧拉角转动变换矩阵计算所述圆盘的运动方程；Calculate the motion equation of the disk based on the Euler angular rotation transformation matrix;
    利用位移插值函数矩阵，计算所述弹性轴的运动方程；Use the displacement interpolation function matrix to calculate the motion equation of the elastic axis;
    基于轴承的轴颈中心坐标，计算所述轴承的运动方程；Calculate the motion equation of the bearing based on the journal center coordinates of the bearing;
    基于所述圆盘的运动方程、所述弹性轴的运动方程以及所述轴承的运动方程，确定所述转子系统的运动方程；Determine an equation of motion of the rotor system based on the equation of motion of the disk, the equation of motion of the elastic shaft, and the equation of motion of the bearing;
    基于所述转子系统的运动方程，计算所述转子系统的临界转速和不平衡响应。Based on the equation of motion of the rotor system, the critical speed and unbalance response of the rotor system are calculated.
  2. 根据权利要求1所述的基于有限元法的转子系统动特性计算方法，其特征在于，所述基于欧拉角转动变换矩阵计算所述圆盘的运动方程包括：The method for calculating dynamic characteristics of a rotor system based on the finite element method according to claim 1, wherein the calculation of the motion equation of the disk based on the Euler angle rotation transformation matrix includes:
    当所述圆盘的轴心与所述圆盘的重心重合时，计算所述圆盘的动能；When the axis of the disk coincides with the center of gravity of the disk, calculate the kinetic energy of the disk;
    基于欧拉角转动变换矩阵以及所述圆盘的动能，计算所述圆盘的运动方程。Based on the Euler angular rotation transformation matrix and the kinetic energy of the disk, the motion equation of the disk is calculated.
  3. 根据权利要求2所述的基于有限元法的转子系统动特性计算方法，其特征在于，所述当所述圆盘的轴心与所述圆盘的重心重合时，计算所述圆盘的动能包括：The method for calculating dynamic characteristics of a rotor system based on the finite element method according to claim 2, characterized in that when the axis of the disk coincides with the center of gravity of the disk, the kinetic energy of the disk is calculated. include:
    当所述圆盘的轴心与所述圆盘的重心重合时，所述圆盘的动能表达式为：When the axis of the disk coincides with the center of gravity of the disk, the kinetic energy expression of the disk is:
    Figure PCTCN2022108728-appb-100001
    Figure PCTCN2022108728-appb-100001
    其中，x、y分别表示所述圆盘的轴心横坐标和纵坐标，
    Figure PCTCN2022108728-appb-100002
    分别表示所述圆盘的轴心横坐标和纵坐标对时间的一阶倒数，m d、J d和J p分别为所述圆盘的质量、所述圆盘的直径转动惯量和所述圆盘的极转动惯量，O 2ξηζ是以所述圆盘的轴心节点为原点，O 2ζ轴与圆盘平面垂直，固结于圆盘的动坐标系，ω ξ，ω η，ω ζ分别表示转速在所述动坐标系的第一分量，第二分量和第三分量；     Among them, x and y respectively represent the abscissa and ordinate of the axis of the disk,
    Figure PCTCN2022108728-appb-100002
    represent the first-order reciprocal of the abscissa and ordinate of the axis of the disk to time respectively, m d , J d and J p are respectively the mass of the disk, the diameter moment of inertia of the disk and the circle The polar moment of inertia of the disk, O 2 ξηζ, is based on the axis node of the disk as the origin. The O 2 ζ axis is perpendicular to the plane of the disk and is fixed in the dynamic coordinate system of the disk, ω ξ , ω η , ω ζ . Represent respectively the first component, the second component and the third component of the rotational speed in the moving coordinate system;
    所述基于欧拉角转动变换矩阵以及所述圆盘的动能，计算所述圆盘的运动方程包括：Calculating the motion equation of the disk based on the Euler angle rotation transformation matrix and the kinetic energy of the disk includes:
    基于所述欧拉角转动变换矩阵，对所述圆盘的动能表达式进行处理，得到处理后的圆盘的动能表达式为：Based on the Euler angle rotation transformation matrix, the kinetic energy expression of the disk is processed, and the processed kinetic energy expression of the disk is obtained:
    Figure PCTCN2022108728-appb-100003
    其中，
    Figure PCTCN2022108728-appb-100004
    Figure PCTCN2022108728-appb-100003
    in,
    Figure PCTCN2022108728-appb-100004
    其中，
    Figure PCTCN2022108728-appb-100005
    分别表示圆盘轴心位移向量的第一分量和第二分量对时间的一阶导数，M d表示圆盘的广义质量，Ω表示圆盘转动角速度，J表示圆盘的转动惯量；     in,
    Figure PCTCN2022108728-appb-100005
    represent the first-order derivative of the first component and the second component of the disk axis displacement vector with respect to time respectively, M d represents the generalized mass of the disk, Ω represents the rotation angular velocity of the disk, and J represents the rotational inertia of the disk;
    根据具有n个自由度的转子系统的状态和位置的Lagrange方程，对处理后的圆盘的动能表达式进行处理，得到所述圆盘的运动方程的表达式为：According to the Lagrange equation of the state and position of the rotor system with n degrees of freedom, the kinetic energy expression of the processed disk is processed, and the expression of the motion equation of the disk is obtained:
    Figure PCTCN2022108728-appb-100006
    Figure PCTCN2022108728-appb-100006
    上式中，{Q 1d}＝[F x，M y] T，{Q 2d}＝[F y，M x] T； In the above formula, {Q 1d } = [F x , M y ] T , {Q 2d } = [F y , M x ] T ;
    其中，
    Figure PCTCN2022108728-appb-100007
    分别表示圆盘轴心位移向量的第一分量和第二分量对时间的二阶导数，Q 1d，Q 2d分别表示法向接触载荷，F x，F y分别表示所述圆盘受到的外力的第一分量和第二分量，M x，M y分别表示力矩的第一分量和第二分量。     in,
    Figure PCTCN2022108728-appb-100007
    represent the second derivative of the first component and the second component of the disk axis displacement vector with respect to time, Q 1d and Q 2d respectively represent the normal contact load, F x and F y respectively represent the external force exerted on the disk. The first component and the second component, M x , My y represent the first component and the second component of the moment respectively.
  4. 根据权利要求1所述的基于有限元法的转子系统动特性计算方法，其特征在于，利用位移插值函数矩阵，计算所述弹性轴的运动方程包括：The method for calculating dynamic characteristics of a rotor system based on the finite element method according to claim 1, characterized in that, using a displacement interpolation function matrix, calculating the motion equation of the elastic axis includes:
    将所述弹性轴分割为多个轴段单元；Divide the elastic shaft into multiple shaft segment units;
    利用位移插值函数矩阵，以及所述轴段单元的节点位移，计算所述轴段单元的动能和弯曲势能；Calculate the kinetic energy and bending potential energy of the shaft segment unit using the displacement interpolation function matrix and the node displacement of the shaft segment unit;
    将所述轴段单元的动能和弯曲势能代入Lagrange方程中，得到所述轴段单元的运动方程。Substituting the kinetic energy and bending potential energy of the shaft segment unit into the Lagrange equation, the motion equation of the shaft segment unit is obtained.
  5. 根据权利要求4所述的基于有限元法的转子系统动特性计算方法，其特征在于，所述利用位移插值函数矩阵，以及所述轴段单元的节点位移，计算所述轴段单元的动能和弯曲势能包括：The method for calculating dynamic characteristics of a rotor system based on the finite element method according to claim 4, characterized in that the displacement interpolation function matrix and the node displacement of the shaft segment unit are used to calculate the kinetic energy sum of the shaft segment unit. Bending potential energy includes:
    利用位移插值函数矩阵，以及所述轴段单元的节点位移，计算所述轴段单元任一截面处的位移的表达式为：Using the displacement interpolation function matrix and the node displacement of the shaft segment unit, the expression for calculating the displacement at any section of the shaft segment unit is:
    Figure PCTCN2022108728-appb-100008
    Figure PCTCN2022108728-appb-100008
    其中，x(z，t)，y(z，t)分别表示所述轴段单元任一截面处的位移，θ x(z，t)，θ y(z，t)表示所述轴段单元任一截面处的角位移，u 1z，u 2z表示直角坐标系下圆盘转度，z表示位置，t表示时间，N为位移插值函数； Among them, x (z, t), y (z, t) respectively represent the displacement at any section of the shaft segment unit, θ x (z, t), θ y (z, t) represent the shaft segment unit The angular displacement at any section, u 1z , u 2z represents the disk rotation under the rectangular coordinate system, z represents the position, t represents the time, and N is the displacement interpolation function;
    基于所述轴段单元任一截面处的位移，计算所述所述轴段单元中任一轴段微元的动能的表达式为：Based on the displacement at any section of the shaft segment unit, the expression for calculating the kinetic energy of any shaft segment element in the shaft segment unit is:
    Figure PCTCN2022108728-appb-100009
    Figure PCTCN2022108728-appb-100009
    其中，dm、j d、j p分别表示所述轴段单元的质量、直径转动惯量和极转动惯量，N’表示轴段单元的2次位移插值多项式； Among them, dm, j d and j p respectively represent the mass, diameter moment of inertia and polar moment of inertia of the shaft segment unit, and N' represents the quadratic displacement interpolation polynomial of the shaft segment unit;
    基于所述轴段单元任一截面处的位移，计算所述所述轴段微元中任意轴段微元的弯曲势能的表达式为：Based on the displacement at any section of the shaft segment unit, the expression for calculating the bending potential energy of any of the shaft segment units is:
    Figure PCTCN2022108728-appb-100010
    Figure PCTCN2022108728-appb-100010
    其中，E为弹性模量，I为横截面对弯曲中性轴的惯性矩，N”表示轴段单元的3次位移插值多项式，
    Figure PCTCN2022108728-appb-100011
    和
    Figure PCTCN2022108728-appb-100012
    分别表示轴段单元任一截面处的沿x、y方向的加速度；     Among them, E is the elastic modulus, I is the moment of inertia of the cross section against the bending neutral axis, and N" represents the third-order displacement interpolation polynomial of the axis segment unit,
    Figure PCTCN2022108728-appb-100011
    and
    Figure PCTCN2022108728-appb-100012
    Respectively represent the acceleration along the x and y directions at any section of the shaft segment unit;
    对于长为l，半径为R的圆形截面轴段单元，将所述轴段微元的动能的表达式以及轴段微元的弯曲势能的表达式沿全长积分，得到所述轴段单元的动能的表达式为：For a circular cross-section shaft segment unit with length l and radius R, the expression of the kinetic energy of the shaft segment element and the expression of the bending potential energy of the shaft segment element are integrated along the entire length to obtain the shaft segment unit. The expression of kinetic energy is:
    Figure PCTCN2022108728-appb-100013
    Figure PCTCN2022108728-appb-100013
    其中，M zT，M zR分别表示表示轴段单元的对角质量矩阵，J z表示轴段单元 的转动惯量； Among them, M zT and M zR respectively represent the diagonal mass matrix representing the axis segment unit, and J z represents the moment of inertia of the axis segment unit;
    所述轴段单元的弯曲势能的表达式为：The expression of the bending potential energy of the shaft segment unit is:
    Figure PCTCN2022108728-appb-100014
    Figure PCTCN2022108728-appb-100014
    其中，K z表示轴段单元的刚度矩阵； Among them, K z represents the stiffness matrix of the axis segment unit;
    将所述轴段单元的动能和弯曲势能代入Lagrange方程中，得到所述轴段单元的运动方程包括：Substituting the kinetic energy and bending potential energy of the shaft segment unit into the Lagrange equation, the motion equation of the shaft segment unit is obtained:
    Figure PCTCN2022108728-appb-100015
    Figure PCTCN2022108728-appb-100015
    式中，[M z]＝[M zR]+[M zT]为对角矩阵，{Q 1z}、{Q 2z}为对应的广义力向量，包括节点处连接的圆盘或相邻轴段的作用力、力矩以及不平衡力。 In the formula, [M z ]=[M zR ]+[M zT ] is a diagonal matrix, {Q 1z }, {Q 2z } are the corresponding generalized force vectors, including the disks or adjacent shaft segments connected at the nodes. forces, moments and unbalanced forces.
  6. 根据权利要求1所述的基于有限元法的转子系统动特性计算方法，其特征在于，所述基于轴承的轴颈中心坐标，计算所述轴承的运动方程的表达式为：The method for calculating the dynamic characteristics of a rotor system based on the finite element method according to claim 1, wherein the expression for calculating the motion equation of the bearing based on the journal center coordinate of the bearing is:
    Figure PCTCN2022108728-appb-100016
    Figure PCTCN2022108728-appb-100016
    其中，M bx，M by分别表示表示轴承在x、y方向上的质量，x b、y b为所述轴承的轴承座中心的坐标，
    Figure PCTCN2022108728-appb-100017
    为所述轴承的轴承座中心的坐标对时间的一阶倒数，
    Figure PCTCN2022108728-appb-100018
    为所述轴承的轴承座中心的坐标对时间的二阶倒数，x s(j)、y s(j)为所述轴承的轴颈中心的坐标，k bxx，k bxy，k byx，k byy分别表示轴承在不同方向上的刚度，c bxx，c bxy，c byx，c byy分别表示轴承在不同方向上 的阻尼。     Among them, M bx and M by respectively represent the mass of the bearing in the x and y directions, x b and y b are the coordinates of the bearing seat center of the bearing,
    Figure PCTCN2022108728-appb-100017
    is the first-order reciprocal of the coordinates of the bearing seat center of the bearing versus time,
    Figure PCTCN2022108728-appb-100018
    is the second-order reciprocal of the coordinates of the bearing seat center of the bearing versus time, x s(j) and y s(j) are the coordinates of the journal center of the bearing, k bxx , k bxy , k byx , k byy Respectively represent the stiffness of the bearing in different directions, c bxx , c bxy , c byx , c byy respectively represent the damping of the bearing in different directions.
  7. 根据权利要求1所述的基于有限元法的转子系统动特性计算方法，其特征在于，所述基于所述圆盘的运动方程、所述弹性轴的运动方程以及所述轴承的运动方程，确定的所述转子系统的运动方程的表达式为：The method for calculating dynamic characteristics of a rotor system based on the finite element method according to claim 1, wherein the equation of motion based on the disk, the equation of motion of the elastic shaft and the equation of motion of the bearing determines The expression of the motion equation of the rotor system is:
    Figure PCTCN2022108728-appb-100019
    Figure PCTCN2022108728-appb-100019
    式中，{U 1}＝[x 1，θ y1，x 2，θ y2，…，x n，θ yn] T、{U 2}＝[y 1，-θ x1，y 2，-θ x2，…，y n，-θ xn] T为转子系统的位移向量，
    Figure PCTCN2022108728-appb-100020
    为转子系统的位移向量对时间的一阶倒数，
    Figure PCTCN2022108728-appb-100021
    为转子系统的位移向量对时间的二阶倒数，[M]、Ω[J]、[K]分别为转子系统的质量矩阵、回转矩阵以及刚度矩阵，都是半带宽为4的2n×2n阶对称稀疏矩阵，{Q 1}、{Q 2}为转子系统的广义力。     In the formula, {U 1 } = [x 1 , θ y1 , x 2 , θ y2 ,..., x n , θ yn ] T , {U 2 } = [y 1 , -θ x1 , y 2 , -θ x2 ,...,y n ,-θ xn ] T is the displacement vector of the rotor system,
    Figure PCTCN2022108728-appb-100020
    is the first-order reciprocal of the displacement vector of the rotor system versus time,
    Figure PCTCN2022108728-appb-100021
    is the second-order reciprocal of the displacement vector of the rotor system versus time, [M], Ω[J], and [K] are the mass matrix, rotation matrix, and stiffness matrix of the rotor system respectively, all of order 2n×2n with a half-bandwidth of 4 Symmetric sparse matrices, {Q 1 }, {Q 2 } are the generalized forces of the rotor system.
  8. 根据权利要求7所述的基于有限元法的转系系统动特性计算方法，其特征在于，所述基于所述转子系统的运动方程，所述转子系统的临界转速和不平衡响应包括：The method for calculating the dynamic characteristics of a rotor system based on the finite element method according to claim 7, characterized in that, based on the motion equation of the rotor system, the critical speed and unbalance response of the rotor system include:
    基于所述转子系统的运动方程，通过微分方程的齐次解，计算当Ω＝ω时转子系统的临界转速；Based on the motion equation of the rotor system, through the homogeneous solution of the differential equation, calculate the critical speed of the rotor system when Ω = ω;
    基于所述转子系统的运动方程，对于各向同性轴承，在不考虑阻尼和轴承座振动的影响，所述转子系统的不平衡响应的表达式为：Based on the motion equation of the rotor system, for isotropic bearings, without considering the effects of damping and bearing seat vibration, the expression of the unbalanced response of the rotor system is:
    Figure PCTCN2022108728-appb-100022
    Figure PCTCN2022108728-appb-100022
    式中，{z}＝{U 1}+{U 2}，{Q}＝{Q 1c}+i{Q 2c}表示不平衡力，i表示虚数，Q 1c和Q 2c分别表示不平衡力在复数平面上分解值。 In the formula, {z}={U 1 }+{U 2 }, {Q}={Q 1c }+i{Q 2c } represents the unbalanced force, i represents the imaginary number, Q 1c and Q 2c represent the unbalanced force respectively. Decompose values on the complex plane.
  9. 一种基于有限元法的转子系统动特性计算设备，其特征在于，包括处理器以及与处理器耦合的通信接口；所述处理器用于运行计算机程序或指令，以实现权利要求1-8任一项所述的基于有限元法的转子系统动特性计算方法。A device for calculating dynamic characteristics of a rotor system based on the finite element method, characterized by including a processor and a communication interface coupled with the processor; the processor is used to run computer programs or instructions to implement any one of claims 1-8 The method for calculating the dynamic characteristics of the rotor system based on the finite element method described in the item.
  10. 一种计算机存储介质，其特征在于，所述计算机存储介质中存储有指令，当所述指令被运行时，实现权利要求1至8任一项所述的基于有限元法的转子系统动特性计算方法。A computer storage medium, characterized in that instructions are stored in the computer storage medium. When the instructions are executed, the calculation of dynamic characteristics of the rotor system based on the finite element method according to any one of claims 1 to 8 is realized. method.
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