CN104361145A

CN104361145A - Rotor dynamics modeling method based on axle closely attached coordinate system

Info

Publication number: CN104361145A
Application number: CN201410483863.9A
Authority: CN
Inventors: 吴锋; 高强; 李明武; 钟万勰
Original assignee: Dalian University of Technology
Current assignee: Dalian University of Technology
Priority date: 2014-09-19
Filing date: 2014-09-19
Publication date: 2015-02-18
Anticipated expiration: 2034-09-19
Also published as: CN104361145B

Abstract

A rotor dynamics modeling method based on an axis-attached coordinate system belongs to the field of rotor dynamics analysis. The invention relates to a rotor dynamics simulation analysis technology, and proposes a dynamic modeling method based on an axis-attached coordinate system. The present invention can be used for rotor structure, such as dynamic simulation modeling analysis of rotor systems such as engines, steam turbines, generators, etc., and can also be applied to related fields such as related experiments, design, inversion analysis, etc., and can be used to calculate the critical speed of rotors , can also be used to investigate the motion stability of the rotor and simulate the motion form of the rotor.

Description

A Rotordynamics Modeling Method Based on Axis-Affixed Coordinate System

技术领域 technical field

本发明涉及转子动力学分析技术，构建了一种贴轴坐标系的转子动力学建模方法。 The invention relates to a rotor dynamics analysis technology, and constructs a rotor dynamics modeling method of an axis-attached coordinate system. the

背景技术 Background technique

旋转机械是现代工业中最重要的动力机械，在电力、交通、航空、化工、能源、军工等行业中有着广泛的应用。然而，有关旋转轴系统的振动模型，现有模型还不够充分，尤其对于非对称轴承、高速旋转机组(例如在汽轮发电机组，工业汽轮机组，喷气发动机等方面)等转子仿真振动的模型，传统模型还需要改进。 Rotating machinery is the most important power machinery in modern industry, and it is widely used in electric power, transportation, aviation, chemical industry, energy, military industry and other industries. However, the existing models are not sufficient for the vibration model of the rotating shaft system, especially for the simulation vibration model of the rotor such as asymmetric bearings and high-speed rotating units (such as in turbogenerators, industrial steam turbines, jet engines, etc.), Traditional models still need improvement. the

本发明可以用于转子结构，如发动机、汽轮机、发电机等转子系统的动力学仿真建模分析，也可应用于相关的试验、设计、反演分析等相关领域，可用于计算转子的临界速度，也可用于考察转子的运动稳定性和仿真转子的运动形态。 The present invention can be used for rotor structure, such as dynamic simulation modeling analysis of rotor systems such as engines, steam turbines, generators, etc., and can also be applied to related fields such as related experiments, design, inversion analysis, etc., and can be used to calculate the critical speed of rotors , can also be used to investigate the motion stability of the rotor and simulate the motion form of the rotor. the

发明内容 Contents of the invention

一种基于贴轴坐标系的转子动力学建模方法，在一根轴承上安装不同尺寸的多个转盘，构成转子系统；选定坐标系，建立转子系统的动力微分方程，求解该动力微分方程，从而进行转子系统的动力学分析；本发明是采用贴轴坐标法建立转子系统的动力微分方程；具体方法如下： A rotor dynamics modeling method based on the axis-attached coordinate system. Multiple turntables of different sizes are installed on a bearing to form a rotor system; the coordinate system is selected to establish the dynamic differential equation of the rotor system and solve the dynamic differential equation , so as to carry out the dynamic analysis of the rotor system; the present invention adopts the axis-attached coordinate method to establish the dynamic differential equation of the rotor system; the specific method is as follows:

(a)以静态时轴承的轴向为z轴方向，z轴的垂直面内建立相互垂直的x轴和y轴，坐标系的原点o位于转盘的圆心，z轴随着轴承旋转，轴承的转角速度即为Ω；将转盘处理为一个等厚度h的实心刚体转盘，质量是m＝ρhπR²，转盘绕轴线z转动惯量为I₁＝mR²/2；转盘绕x轴或y轴转动惯量I₂＝mR²/4；转盘的圆心位于z轴上；转子系统运动时，设转盘圆心产生线位移为q_x,q_y；转盘围绕x轴或y轴的转角为θ_x,θ_y； (a) Take the axial direction of the bearing at static state as the z-axis direction, establish the x-axis and y-axis perpendicular to each other in the vertical plane of the z-axis, the origin o of the coordinate system is located at the center of the turntable, the z-axis rotates with the bearing, and the bearing's The rotational angular velocity is Ω; the turntable is treated as a solid rigid turntable with equal thickness h, the mass is m=phπR ² , the moment of inertia of the turntable around the axis z is I ₁ =mR ² /2; the moment of inertia of the turntable around the x-axis or y-axis I ₂ ＝mR ² /4; the center of the turntable is on the z-axis; when the rotor system moves, set the line displacement generated by the center of the turntable as q _x , q _y ; the rotation angle of the turntable around the x-axis or y-axis is θ _x , θ _y ;

按Timoshenco梁理论建立轴承的刚度矩阵K_b，位移q_x,q_y和θ_x,θ_y是独立的； The stiffness matrix K _b of the bearing is established according to the Timoshenco beam theory, and the displacements q _x , q _y and θ _x , θ _y are independent;

(b)根据转盘质心的相对位移q_x,q_y和转轴的转动角速度Ω，建立起其绝对线速度表达式： (b) According to the relative displacement q _x , q _y of the center of mass of the turntable and the rotational angular velocity Ω of the rotating shaft, the expression of its absolute linear velocity is established:

(c)在贴轴坐标内计算转盘平动的线速度动能T_q； (c) Calculate the linear velocity kinetic energy T _q of the translational motion of the turntable in the axis-attached coordinates;

(d)由于转盘围绕x轴或y轴的转角θ_x(t),θ_y(t)是小变形，因此采用角位移向量θ(t)＝{θ_x θ_y}^T描述转盘围绕x轴或y轴的转角，利用向量的特性，可以给出转盘围绕x轴或y轴转角的绝对角速度向量为 (d) Since the rotation angle θ _x (t) and θ _y (t) of the turntable around the x-axis or y-axis are small deformations, the angular displacement vector θ(t)={θ _x θ _y } ^T is used to describe the rotation of the turntable around the x-axis Or the rotation angle of the y-axis, using the characteristics of the vector, the absolute angular velocity vector of the turntable around the x-axis or y-axis rotation angle can be given as

$\overset{. .}{θ θ} ((t t)) + + Ω Ω \times \times θ θ = = (({\overset{. .}{θ θ}}_{y the y} + + Ω Ω {θ θ}_{x x})) {i i}_{y the y} + + (({\overset{. .}{θ θ}}_{x x} - - Ω Ω {θ θ}_{y the y})) {i i}_{x x}$

(e)根据转盘围绕x轴或y轴转角的绝对角速度向量，计算转盘的转动动能： (e) According to the absolute angular velocity vector of the rotation angle of the turntable around the x-axis or y-axis, calculate the rotational kinetic energy of the turntable:

$\begin{matrix} {T T}_{θ θ} = = {I I}_{22} {(({\overset{. .}{θ θ}}_{x x} - - Ω Ω {θ θ}_{y the y}))}^{22} / / 22 + + {I I}_{22} {(({\overset{. .}{θ θ}}_{y the y} + + Ω Ω {θ θ}_{x x}))}^{22} / / 22 = = \\ {I I}_{22} (({\overset{. .}{θ θ}}_{x x}^{22} + + {\overset{. .}{θ θ}}_{y the y}^{22})) / / 22 + + {I I}_{22} Ω Ω (({\overset{. .}{θ θ}}_{y the y} {θ θ}_{x x} - - {\overset{. .}{θ θ}}_{x x} {θ θ}_{y the y})) + + {I I}_{22} {Ω Ω}^{22} (({θ θ}_{x x}^{22} + + {θ θ}_{y the y}^{22})) / / 22 \end{matrix}$

其中I₂＝mR²/4，是转盘围绕x轴或y轴的转动惯量； Among them, I ₂ =mR ² /4, which is the moment of inertia of the turntable around the x-axis or y-axis;

(f)修正步骤(e)的转动动能；由于转轴的高速旋转角速度为Ω，转轴会因转盘围绕x轴或y轴的转角θ_x(t),θ_y(t)而有方向变化，导致转盘偏斜了，因此转动惯量修正为：利用转动惯量，把轴向转动的能量I₁Ω²/2要修改为： (f) correct the rotational kinetic energy of step (e); since the high-speed rotational angular velocity of the rotating shaft is Ω, the rotating shaft will have a direction change due to the rotation angle θ _x (t) and θ _y (t) of the turntable around the x-axis or y-axis, resulting in The turntable is deflected, so the moment of inertia is corrected as: Using the moment of inertia, the energy I ₁ Ω ² /2 of axial rotation should be modified as:

(g)贴轴坐标内转盘的角位移θ_x(t),θ_y(t)与Timoshenco梁的转角ψ_x,ψ_y有对应关系如下式： (g) The angular displacement θ _x (t), θ _y (t) of the turntable in the axis-attached coordinates has a corresponding relationship with the rotation angle ψ _x , ψ _y of the Timoshenco beam as follows:

θ_y＝ψ_x,θ_x＝-ψ_y θ _y = ψ _x , θ _x = -ψ _y

把转盘(2)动能中的变量θ_x(t),θ_y(t)用Timoshenco梁的转角ψ_x,ψ_y表示，则有： The variables θ _x (t), θ _y (t) in the kinetic energy of the turntable (2) are represented by the rotation angle ψ _x , ψ _y of the Timoshenco beam, then:

$T T = = \frac{{\overset{. .}{q q}}^{T T}}{22} {m m}_{i i} \overset{. .}{q q} + + {\overset{. .}{q q}}^{T T} Ω Ω {g g}_{i i} q q + + \frac{{q q}^{T T}}{22} {Ω Ω}^{22} {k k}_{Ti Ti} q q$

其中 in

q＝{q_x q_y ψ_x ψ_y}^T q＝{q _x q _y ψ _x ψ _y } ^T

(h)根据步骤(g)中的动能表达式，可以得到转子动力学在贴轴坐标系下的振动模型为： (h) According to the kinetic energy expression in step (g), the vibration model of the rotor dynamics in the axis-attached coordinate system can be obtained as:

$M m \overset{. . . .}{q q} + + ΩG ΩG \overset{. .}{q q} + + {K K}_{s the s} q q = = 00$

根据该振动模型，从而可以进行转子系统的动力学分析。 Based on the vibration model, dynamic analysis of the rotor system can be performed. the

所述一种基于贴轴坐标系的转子动力学建模方法，其特征在于步骤(c)中，线性位移动能表达式为： The rotor dynamics modeling method based on the axis-attached coordinate system is characterized in that in step (c), the linear displacement kinetic energy expression is:

$\begin{matrix} {T T}_{q q} = = m m {(({\overset{. .}{q q}}_{x x} - - Ω Ω {q q}_{y the y}))}^{22} / / 22 + + m m {(({\overset{. .}{q q}}_{y the y} + + {Ωq Ωq}_{x x}))}^{22} / / 22 = = \\ m m (({\overset{. .}{q q}}_{x x}^{22} + + {\overset{. .}{q q}}_{y the y}^{22})) / / 22 + + mΩ mΩ (({\overset{. .}{q q}}_{y the y} {q q}_{x x} - - {\overset{. .}{q q}}_{x x} {q q}_{y the y})) + + m m {Ω Ω}^{22} (({q q}_{x x}^{22} + + {q q}_{y the y}^{22})) / / 22 \end{matrix}$

所述一种基于贴轴坐标系的转子动力学建模方法，其特征在于步骤(f)中动能表达式为： The rotor dynamics modeling method based on the axis-attached coordinate system is characterized in that the kinetic energy expression in the step (f) is:

$\begin{matrix} {T T}_{θ θ - -} = = {I I}_{22} {(({\overset{. .}{θ θ}}_{x x} - - Ω Ω {θ θ}_{y the y}))}^{22} / / 22 + + {I I}_{22} {(({\overset{. .}{θ θ}}_{y the y} + + Ω Ω {θ θ}_{x x}))}^{22} / / 22 = = \\ {I I}_{22} (({\overset{. .}{θ θ}}_{x x}^{22} + + {\overset{. .}{θ θ}}_{y the y}^{22})) / / 22 + + {I I}_{22} Ω Ω (({\overset{. .}{θ θ}}_{y the y} {θ θ}_{x x} - - {\overset{. .}{θ θ}}_{x x} {θ θ}_{y the y})) + + {I I}_{22} {Ω Ω}^{22} (({θ θ}_{x x}^{22} + + {θ θ}_{y the y}^{22})) / / 22 - - {I I}_{11} {Ω Ω}^{22} (({θ θ}_{x x}^{22} + + {θ θ}_{y the y}^{22})) / / 22 \end{matrix}$

所述一种基于贴轴坐标系的转子动力学建模方法，其特征在于步骤(g)中各矩阵表达式为： Described a kind of rotor dynamics modeling method based on the axis-attached coordinate system is characterized in that each matrix expression in step (g) is:

${m m}_{i i} = = [\begin{matrix} m m & 00 \\ 00 & m m \\ {I I}_{22} & 00 \\ 00 & {I I}_{22} \end{matrix}],, {g g}_{i i} = = [\begin{matrix} 00 & - - m m \\ m m & 00 \\ 00 & - - {I I}_{22} \\ {I I}_{22} & 00 \end{matrix}],, {k k}_{Ti Ti} = = [\begin{matrix} m m & 00 \\ 00 & m m \\ - - {I I}_{22} & 00 \\ 00 & - - {I I}_{22} \end{matrix}]$

所述一种基于贴轴坐标系的转子动力学建模方法，其特征在于步骤(i)中，各矩阵表达式为： The rotor dynamics modeling method based on the axis-attached coordinate system is characterized in that in step (i), each matrix expression is:

K_s＝K_b-Ω²K_T K _s =K _b -Ω ² K _T

M＝diag{m₁,m₂,…,m_N},K_T＝diag{k_T1,k_T2,…,k_TN} M=diag{m ₁ ,m ₂ ,...,m _N }, K _T ＝diag{k _T1 ,k _T2 ,...,k _TN }

$G G = = diag diag {{{G G}_{11},, {G G}_{22},, . . . . . .,, {G G}_{N N}}},, {G G}_{i i} = = {g g}_{i i} - - {g g}_{i i}^{T T}$

本发明的有益积极效果： Beneficial positive effect of the present invention:

本发明基于贴轴坐标建立的转子动力学方程，较传统模型相比，在步骤(g)中同时考虑了平动和转角的陀螺力，且因为是在贴轴坐标下建立的模型，因此易于与轴承的弹性变形相联系。本发明可以用于转子结构，如发动机、汽轮机、发电机等转子系统的动力学仿真建模分析，也可应用于相关的试验、设计、反演分析等相关领域，可用于计算转子的临界速度，也可用于考察转子的运动稳定性和仿真转子的运动形态。 Compared with the traditional model, the rotor dynamics equation established based on the axis-attached coordinates of the present invention considers the gyroscopic force of translation and rotation angle simultaneously in step (g), and because it is a model established under the axis-attached coordinates, it is easy to Linked to the elastic deformation of the bearing. The present invention can be used for rotor structure, such as dynamic simulation modeling analysis of rotor systems such as engines, steam turbines, generators, etc., and can also be applied to related fields such as related experiments, design, inversion analysis, etc., and can be used to calculate the critical speed of rotors , can also be used to investigate the motion stability of the rotor and simulate the motion form of the rotor. the

附图说明 Description of drawings

图1是一个转盘静态转子系统。 Figure 1 is a turntable static rotor system. the

图2是变形后一个转盘转子系统，其中θ_x,θ_y分别为转盘围绕x轴和y轴的转动角度，q_x,q_y是转盘圆心的位移。 Figure 2 shows a turntable rotor system after deformation, where θ _x , θ _y are the rotation angles of the turntable around the x-axis and y-axis respectively, and q _x , q _y are the displacements of the center of the turntable.

图中：1.轴承，2.转盘。 In the figure: 1. bearing, 2. turntable. the

具体实施方式 Detailed ways

下面结合附图阐述本发明的实施方式： Set forth the embodiment of the present invention below in conjunction with accompanying drawing:

在轴承(1)上安装不同尺寸的多个转盘(2)，构成转子系统， Multiple turntables (2) of different sizes are installed on the bearing (1) to form a rotor system,

(a)对多个转盘(2)，分别计算其质量m_i和转动惯量 (a) For multiple turntables (2), calculate their mass m _i and moment of inertia respectively

(b)按Timoshenco梁理论建立轴承的刚度矩阵K_b，K_b可以利用有限元软件：大连理工大学自主研发的SIPESC，SIPESC软件是大连理工大学工程力学系开发的工程计算分析软件平台，其功能包括集成开发环境、面向系统集成的活动流程图定制、工程数据库管理系统、开放式结构有限元分析系统、集成优化计算系统等，其中有限元分析系统中集成了方程求解模块、有限元后处理模块等，其有限元模块中包含了Timoshenco梁的刚度矩阵。 (b) Establish the stiffness matrix K _b of the bearing according to Timoshenco beam theory, K _b can use finite element software: SIPESC independently developed by Dalian University of Technology, SIPESC software is an engineering calculation and analysis software platform developed by the Department of Engineering Mechanics of Dalian University of Technology, its function Including integrated development environment, activity flow chart customization for system integration, engineering database management system, open structure finite element analysis system, integrated optimization calculation system, etc., among which the finite element analysis system integrates equation solving module and finite element post-processing module etc., the stiffness matrix of the Timoshenco beam is included in the finite element module.

(c)计算第i个转盘(2)的线速度动能所需矩阵： (c) Calculate the matrix required for the linear velocity kinetic energy of the i-th turntable (2):

${m m}_{i i} = = [\begin{matrix} {m m}_{i i} & 00 \\ 00 & {m m}_{i i} \\ {I I}_{22 i i} & 00 \\ 00 & {I I}_{22 i i} \end{matrix}],, {g g}_{i i} = = [\begin{matrix} 00 & - - {m m}_{i i} \\ {m m}_{i i} & 00 \\ 00 & - - {I I}_{22 i i} \\ {I I}_{22 i i} & 00 \end{matrix}],, {k k}_{Ti Ti} = = [\begin{matrix} {m m}_{i i} & 00 \\ 00 & {m m}_{i i} \\ - - {I I}_{22 i i} & 00 \\ 00 & - - {I I}_{22 i i} \end{matrix}]$

(d)建立转子动力学在贴轴坐标系下的振动模型为： (d) Establish the vibration model of the rotor dynamics in the axis-attached coordinate system as:

其中， in,

K_s＝K_b-Ω²K_T K _s =K _b -Ω ² K _T

$G G = = diag diag {{{G G}_{11},, {G G}_{22},, . . . . . .,, {G G}_{N N}}},, {G G}_{i i} = = {g g}_{i i} - - {g g}_{i i}^{T T} . .$

Claims

1. A rotor dynamics modeling method based on a shaft-attached coordinate system is characterized in that a plurality of rotary tables (2) with different sizes are mounted on a bearing (1) to form a rotor system; selecting a coordinate system, establishing a dynamic differential equation of the rotor system, and solving the dynamic differential equation so as to perform dynamic analysis on the rotor system; the method is characterized in that a dynamic differential equation of a rotor system is established by adopting a shaft-pasting coordinate method; the specific method comprises the following steps:

(a) the axial direction of the bearing (1) in a static state is taken as the direction of a z axis, an x axis and a y axis which are perpendicular to each other are established in the vertical plane of the z axis, and the origin o of a coordinate system is positionedThe center of the circle of the turntable (2) and the z axis rotate along with the bearing (1), and the angular velocity of the bearing (1) is omega; the turntable (2) is processed into a solid rigid turntable with the same thickness h, and the mass is m ═ rho pi R²The inertia moment of the turntable (2) about the axis z is I₁＝mR²2; the rotary inertia I of the rotary disc (2) around the x axis or the y axis₂＝mR²(ii)/4; the circle center of the turntable (2) is positioned on the z axis; when the rotor system moves, the linear displacement generated by the circle center of the rotary table (2) is set as q_x,q_y(ii) a The rotation angle of the rotating disc (2) around the x axis or the y axis is theta_x,θ_y；

Establishing a rigidity matrix K of the bearing (1) according to the Timoshenco beam theory_bOf displacement q_x,q_yAnd theta_x,θ_yIs independent;

(b) according to the relative displacement q of the center of mass of the rotating disc (2)_x,q_yAnd the rotation angular speed omega of the rotating shaft (1), and an absolute linear speed expression of the rotation angular speed omega is established:

(c) calculating the linear velocity kinetic energy T of the translation of the turntable (2) in the sticking axis coordinate_q；

(d) Due to the rotation angle theta of the rotating disc (2) around the x-axis or the y-axis_x(t),θ_y(t) is a small deformation, so an angular displacement vector θ (t) { θ ═ is used_x θ_y}^TDescribing the rotation angle of the rotary table (2) around the x-axis or the y-axis, the absolute angular velocity vector of the rotation angle of the rotary table (2) around the x-axis or the y-axis can be given as

(e) Calculating the rotational kinetic energy of the rotating disc (2) according to the absolute angular velocity vector of the rotating disc (2) around the x-axis or the y-axis:

wherein I₂＝mR²-4, the moment of inertia of the turntable (2) about the x-axis or the y-axis;

(f) modifying the rotational kinetic energy of step (e); because the high-speed rotation angular velocity of the rotating shaft (1) is omega, the rotating shaft (1) can rotate at an angle theta around the x axis or the y axis due to the rotating disc (2)_x(t),θ_y(t) there is a change in direction, which causes the turntable (2) to deflect, so that the moment of inertia is corrected to:using the moment of inertia to turn the axial energy I₁Ω²The/2 modification is:

(g) angular displacement theta of inner rotating disc (2) in axial coordinate_x(t),θ_y(t) angle of rotation phi with Timoshenco Beam_x,ψ_yThere is a corresponding relationship as follows:

θ_y＝ψ_x,θ_x＝-ψ_y

the variable theta in the kinetic energy of the rotary disc (2)_x(t),θ_y(t) corner psi with Timoshenco beams_x,ψ_yThis means that there are:

wherein

q＝{q_x q_y ψ_x ψ_y}^T

(h) According to the kinetic energy expression in the step (g), a vibration model of rotor dynamics in a paraxial coordinate system can be obtained as follows:

from this vibration model, a kinetic analysis of the rotor system can thus be performed.

2. The method according to claim 1, wherein in step (c), the linear displacement kinetic energy expression is:

3. the method according to claim 1, wherein the kinetic energy expression in step (f) is as follows:

4. the method according to claim 1, wherein the matrix expressions in step (g) are as follows:

m_{i} = [\begin{matrix} m & 0 \\ 0 & m \\ I_{2} & 0 \\ 0 & I_{2} \end{matrix}], g_{i} = [\begin{matrix} 0 & - m \\ m & 0 \\ 0 & - I_{2} \\ I_{2} & 0 \end{matrix}], k_{Ti} = [\begin{matrix} m & 0 \\ 0 & m \\ - I_{2} & 0 \\ 0 & - I_{2} \end{matrix}] .

5. the method according to claim 1, wherein in step (i), each matrix expression is as follows:

K_s＝K_b-Ω²K_T

M＝diag{m₁,m₂,…,m_N},K_T＝diag{k_T1,k_T2,…,k_TN}

G = diag {G_{1}, G_{2}, . . ., G_{N}}, G_{i} = g_{i} - g_{i}^{T} .