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

The invention discloses a rotor dynamics modeling method based on an axle closely attached coordinate system, belongs to the field of rotor dynamics analysis, relates to the technology of rotor dynamics simulation analysis, and discloses the rotor dynamics modeling method based on the axle closely attached coordinate system. The rotor dynamics modeling method can be used for dynamics simulation modeling analysis of rotor structures such as an engine, a steam turbine, an electric generator and other rotor systems, calculating the critical speed of a rotor, investigating the kinetic stability of the rotor and simulating the motion morphology of the rotor, and can also be applicable to related tests, designs, inverse analysis and other related fields.

Description

rotor dynamics modeling method based on axial coordinate system

Technical Field

The invention relates to a rotor dynamics analysis technology, and discloses a rotor dynamics modeling method of a shaft-attached coordinate system.

Background

The rotary machine is the most important power machine in modern industry, and has wide application in the industries of electric power, traffic, aviation, chemical industry, energy, military industry and the like. However, the existing model is not sufficient for the vibration model of the rotating shaft system, and especially for the model of the simulation vibration of the rotor of the asymmetric bearing, the high-speed rotating unit (for example, in the aspects of a turbo generator unit, an industrial steam turbine unit, a jet engine and the like), and the like, the traditional model needs to be improved.

The invention can be used for the dynamic simulation modeling analysis of rotor structures, such as rotor systems of engines, steam turbines, generators and the like, can also be applied to relevant fields of relevant tests, designs, inversion analysis and the like, can be used for calculating the critical speed of the rotor, and can also be used for inspecting the motion stability of the rotor and simulating the motion form of the rotor.

Disclosure of Invention

A rotor dynamics modeling method based on a shaft-attached coordinate system is characterized in that a plurality of turntables with different sizes are mounted on a bearing 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 invention adopts a shaft-fitting coordinate method to establish a dynamic differential equation of a rotor system; the specific method comprises the following steps:

(a) the axial direction of the bearing 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, the origin o of a coordinate system is positioned at the center of a circle of a turntable, the z axis rotates along with the bearing, and the rotating angular speed of the bearing is omega; processing the turntable into a solid rigid turntable with the same thickness h, wherein the mass is m ═ rho h pi R²The inertia of the turntable about the axis z is I₁＝mR²2; moment of inertia I of revolution about x-axis or y-axis₂＝mR²(ii)/4; the circle center of the rotary table is positioned on the z axis; when the rotor system moves, the center of the rotary table generates linear displacement of q_x,q_y(ii) a The angle of rotation of the turntable about the x-or y-axis being theta_x,θ_y；

Establishing rigidity matrix K of bearing according to Timoshenco beam theory_bOf displacement q_x,q_yAnd theta_x,θ_yIs independent;

(b) relative displacement q according to the center of mass of the turntable_x,q_yAnd the rotation angular velocity omega of the rotating shaft to establish the absolute value thereofLinear velocity expression:

(c) calculating linear velocity kinetic energy T of turntable translation in shaft pasting coordinates_q；

(d) Due to the angle of rotation theta of the turntable about the x-axis or 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 turntable around the x-axis or the y-axis, by using the characteristics of the vector, the absolute angular velocity vector of the rotation angle of the turntable around the x-axis or the y-axis can be given as

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

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

(f) modifying the rotational kinetic energy of step (e); because the high-speed rotation angular velocity of the rotating shaft is omega, the rotating shaft can rotate around the rotation angle theta of the x-axis or the y-axis due to the rotating disc_x(t),θ_y(t) there is a change in direction that causes the turntable to deflect, so 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 rotating disc 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.

The rotor dynamics modeling method based on the axial coordinate system is characterized in that in the step (c), the linear displacement kinetic energy expression is as follows:

the rotor dynamics modeling method based on the axial coordinate system is characterized in that the kinetic energy expression in the step (f) is as follows:

the rotor dynamics modeling method based on the axial coordinate system is characterized in that in the step (g), each matrix expression is 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}]

the rotor dynamics modeling method based on the axial coordinate system is characterized in that in the 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}

The invention has the following beneficial and positive effects:

compared with the traditional model, the rotor dynamics equation established based on the axial coordinate in the invention considers the gyroscopic forces of translation and rotation angle in step (g) at the same time, and is easy to be associated with the elastic deformation of the bearing because of the model established under the axial coordinate. The invention can be used for the dynamic simulation modeling analysis of rotor structures, such as rotor systems of engines, steam turbines, generators and the like, can also be applied to relevant fields of relevant tests, designs, inversion analysis and the like, can be used for calculating the critical speed of the rotor, and can also be used for inspecting the motion stability of the rotor and simulating the motion form of the rotor.

Drawings

FIG. 1 is a rotary disk static rotor system.

FIG. 2 is a modified rotor system of a rotary disk, where θ_x,θ_yThe rotation angles of the turntable about the x-axis and the y-axis, q_x,q_yIs the displacement of the center of the circle of the turntable.

In the figure: 1. bearing, 2. turntable.

Detailed Description

Embodiments of the invention are described below with reference to the accompanying drawings:

a plurality of turntables (2) with different sizes are arranged on the bearing (1) to form a rotor system,

(a) for a plurality of turntables (2), the mass m is calculated_iAnd moment of inertia

(b) Establishing rigidity matrix K of bearing according to Timoshenco beam theory_b，K_bFinite element software may be utilized: the SIPSCs are independently developed by university of general technology, SIPSCs software is an engineering calculation analysis software platform developed by the university of general technology engineering department, and the functions of the SIPSCs comprise an integrated development environment, system integration-oriented activity flow chart customization, an engineering database management system, an open structure finite element analysis system, an integrated optimization calculation system and the like, wherein an equation solving module, a finite element post-processing module and the like are integrated in the finite element analysis system, and the finite element module comprises a rigidity matrix of a Timoshenco beam.

(c) Calculating a matrix required by linear velocity kinetic energy of the ith rotating disc (2):

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

(d) the vibration model of rotor dynamics under a close-to-axis coordinate system is established as follows:

wherein,

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} .

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

wherein

q＝{q_x q_y ψ_x ψ_y}^T

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} .