A cross-coordinate-system dynamic modeling method for a misaligned rotor with a skew angle

Abstract

The application discloses a kind of cross-coordinate system dynamics modeling methods of angle misalignment fault rotor, comprising: according to the axis feature of rotor, the rotor that occurs angle misalignment fault is divided into connection substructure, left rotor substructure, right rotor substructure, wherein, left rotor substructure and connection substructure, right rotor substructure are located in different coordinate system respectively;In respective coordinate system, the motion differential equation of left rotor substructure, right rotor substructure is established;According to the node position of connection substructure, the motion differential equation of left rotor substructure, right rotor substructure and the cross-coordinate system stiffness equation of connection substructure are grouped, and the whole rotor cross-coordinate system motion differential equation is formed.The rotor model established by the modeling method of the application can accurately simulate the axis feature after rotor angle misalignment fault and is simple and fast.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of mechanics, and particularly relates to a cross-coordinate-system dynamic modeling method for a misaligned-angel fault rotor. BACKGROUND

[0002] In rotor equipment such as an aero-engine and a steam turbine, misalignment faults of rotors are common, which can cause complex rotor vibration problems. Therefore, modeling and simulation of misaligned rotors have significant meaning for dynamic prediction of rotor equipment. In existing finite element models of misaligned rotors, the angle between the axes of each part of the rotor after a misalignment fault occurs is almost ignored, and the axis of the entire rotor is simplified as a straight line. This modeling method in which the axes coincide cannot accurately describe the misalignment fault characteristics, and is only an approximate simulation method. In view of this problem, a document (Gupta T C, Gupta K. Modeling of Flexible Coupling to Connect Misaligned Flexible Rotors Supported on Ball Bearings [C] / / Asme Turbo Expo: Turbine Technical Conference & Exposition. American Society of Mechanical Engineers, 2014.) establishes a finite element model of a misaligned-angel rotor by using a beam element, in which the rotors on both sides of the connecting structure are located in different coordinate systems, as shown in FIG. 1. When modeling, coordinate transformation is performed on all degrees of freedom of the right rotor in the X2Y2Z2 coordinate system, and after the transformation, the degrees of freedom of the entire rotor are in the X1Y1Z1 coordinate system. However, the coordinate transformation needs to be performed on all degrees of freedom of the right rotor, which increases the modeling difficulty of the model. Therefore, there is an urgent need for a cross-coordinate-system dynamic modeling method for a misaligned-angel fault rotor to solve the above problems. Figure 1 SUMMARY

[0003] The application aims to provide a cross-coordinate-system dynamic modeling method for a misaligned-angel fault rotor to solve the problems that in existing finite element models of misaligned rotors, the angle between the axes of each part of the rotor after a misalignment fault occurs is almost ignored, and the axis of the entire rotor is simplified as a straight line, and the cross-coordinate-system dynamic model of a misaligned-angel fault rotor is complex and cumbersome.

[0004] The application provides a cross-coordinate-system dynamic modeling method for a misaligned-angel fault rotor, which comprises the following steps:

[0005] ​According to the axis feature of the rotor, the rotor with the eccentricity misalignment fault is divided into a connecting substructure, a left rotor substructure and a right rotor substructure, wherein the left rotor substructure and the connecting substructure and the right rotor substructure are located in different coordinate systems respectively;

[0006] The motion differential equations of the left rotor substructure and the right rotor substructure are established in the respective coordinate systems;

[0007] The boundary nodes of the connecting substructure are subjected to coordinate transformation, and the cross-coordinate-system stiffness equation of the connecting substructure is established;

[0008] According to the node positions of the connecting substructure, the motion differential equations of the left rotor substructure and the right rotor substructure and the cross-coordinate-system stiffness equation of the connecting substructure are grouped to form the entire rotor cross-coordinate-system motion differential equation.

[0009] Further, in the step of dividing the rotor with the eccentricity misalignment fault into the connecting substructure, the left rotor substructure and the right rotor substructure according to the axis feature of the rotor, the left rotor substructure comprises nodes 1 to 4, the right rotor substructure comprises nodes 5 to 8, the connecting substructure mass is concentrated into nodes 4 and 5, the left rotor substructure and the connecting substructure are located in a coordinate system O1X1Y1Z1, the right rotor substructure is located in a coordinate system O2X2Y2Z2, the two parts of the rotor rotate around the Y1 and Y2 axes at a rotational speed ω, the coordinate system O2X2Y2Z2 is obtained by rotating the coordinate system O1X1Y1Z1 around the X1 axis by α and then translating to the right, the rotational freedom of the rotor around the axis is ignored, and a coordinate transformation matrix formula 1 from the coordinate system O1X1Y1Z1 to the coordinate system O2X2Y2Z2 is obtained:

[0010] Further, in the step of establishing the motion differential equations of the left rotor substructure and the right rotor substructure in the respective coordinate systems, the motion differential equation of the right rotor substructure is established by the following method:

[0011] The generalized coordinates of the right rotor substructure are formula 2:

[0012] The right rotor substructure is located in the coordinate system O2X2Y2Z2 and rotates around the Y2 axis at a rotational speed ω, and the kinetic energy of the right rotor substructure is formula 3: wherein,

[0013] m i =diag{m i m i m i J di Jdi};

[0014] Potential energy of right rotor substructure 4:

[0015] wherein and are stiffness matrices of the shaft segment between node 5-node 6, node 6-node 7 and node 7-node 8 in the coordinate system O2X2Y2Z2, are stiffness matrices of the bearings at node 6, node 8 in the coordinate system O2X2Y2Z2;

[0016] Substitute equation 3 and equation 4 into Lagrange equation, get the motion differential equation of right rotor substructure 5:

[0017] wherein wherein

[0018] wherein is the additional load, including the misalignment load generated by the state change of the connecting substructure.

[0019] Further, in the step of establishing the motion differential equation of the left rotor substructure and the right rotor substructure in the respective coordinate system, the method for establishing the motion differential equation of the left rotor substructure is as follows:

[0020] The generalized coordinates of the left rotor substructure are as follows:

[0021] Using the same method as that for obtaining the motion differential equation of the right rotor substructure, the motion differential equation of the left rotor substructure is obtained as equation 7 wherein

[0022] Further, the coordinate transformation of the boundary nodes of the connecting substructure is performed to establish the stiffness equation of the connecting substructure across the coordinate system, which includes that the connecting substructure has an initial deformation 8: The coordinates of the right side nodes of the connecting substructure and the left side nodes belong to different coordinate systems, and the relative deformation of the two sides of the connecting substructure is obtained through coordinate transformation 9: wherein is the transformation matrix across the coordinate system, is the set of mixed coordinate vectors of the two boundary nodes;

[0023] Potential energy of the connecting substructure in its own coordinate system k 10 wherein is the stiffness matrix of the connecting substructure in the coordinate system k;

[0024] Substituting equation 9 into equation 10, equation 11 is obtained:

[0025]

[0026] Substituting the potential energy of the connecting substructure into the Lagrange equation, equation 12 is obtained

[0027] wherein is the cross-coordinate system stiffness matrix of the connecting substructure.

[0028] Further, according to the node positions of the connecting substructure, the motion differential equations of the left and right rotor substructures and the cross-coordinate system stiffness equation of the connecting substructure are grouped to form the entire rotor cross-coordinate system motion differential equation, including:

[0029] According to the coordinate systems of the two nodes, the cross-coordinate system stiffness matrix of the connecting substructure is blocked, as shown in equation 13:

[0030]

[0031] Equation 13 can be rewritten as equation 14,

[0032]

[0033] The stiffness matrices and load vectors of the left and right rotors are blocked respectively, and J is the boundary node degree of freedom, and I is the degree of freedom except the boundary node; finally, the motion differential equations of the left and right rotors and the stiffness equation of the connecting substructure are grouped according to the corresponding nodes to obtain the complete rotor cross-coordinate system motion differential equation 15:

[0034]

[0035] wherein:

[0036]

[0037]

[0038] wherein:

[0039]

[0040]

[0041] The beneficial effects of the present application are as follows: the present application provides a cross-coordinate system dynamic modeling method for a rotor with a misalignment fault of a deflection angle, on the one hand, the rotor model established by the modeling method of the present application can accurately simulate the axis characteristics of the rotor after the misalignment fault of the deflection angle, on the other hand, compared with the model established in the prior art, the modeling method of the present application does not need to perform coordinate transformation on the motion differential equation of the rotor on one side of the connecting substructure, thereby greatly simplifying the modeling process, in addition, only the coordinate transformation is performed on the connecting substructure, and the rotor substructures on both sides are still located in the physical space, and the degree of freedom reduction processing can be performed on the rotor substructures. BRIEF DESCRIPTION OF DRAWINGS

[0042] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments will be briefly introduced as follows, obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without any creative effort on the basis of these drawings.

[0043] Figure 1 The axis characteristic diagram of the misalignment rotor of the cross-coordinate system dynamic modeling method for a rotor with a misalignment fault of a deflection angle;

[0044] Figure 2 The modeling flowchart of the cross-coordinate system dynamic modeling method for a rotor with a misalignment fault of a deflection angle;

[0045] Figure 3 The rotor model with a misalignment fault of a deflection angle of the cross-coordinate system dynamic modeling method for a rotor with a misalignment fault of a deflection angle;

[0046] Figure 4 The substructure division method of the rotor model of the cross-coordinate system dynamic modeling method for a rotor with a misalignment fault of a deflection angle. DETAILED DESCRIPTION

[0047] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict. The present application will be described in detail below with reference to the drawings and in combination with the embodiments. It should be pointed out that the following detailed description is exemplary and is intended to provide further description of the present application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as generally understood by those skilled in the art.

[0048] Please refer to Figure 2 The present application provides a cross-coordinate system dynamic modeling method for a rotor with a misalignment fault of a deflection angle, including the following steps:

[0049] Step S101: Based on the axial characteristics of the rotor, the rotor that has an angular misalignment fault is divided into a connecting substructure, a left rotor substructure, and a right rotor substructure. The left rotor substructure, the connecting substructure, and the right rotor substructure are located in different coordinate systems.

[0050] Figure 3 The present invention relates to a cross-coordinate system dynamic modeling method for a rotor with an angular misalignment fault, wherein the rotor axes on both sides of the connecting substructure have an included angle.

[0051] Figure 4 A substructure partitioning method is proposed for cross-coordinate system dynamic modeling of a rotor with misalignment fault. The left rotor portion of the connecting substructure is divided into a left rotor substructure, including nodes 1-4; the right rotor portion of the connecting substructure is divided into a right rotor substructure, including nodes 5-8. The mass of the connecting structure is concentrated in nodes 4 and 5, and its mass is reflected in both substructures. The connecting substructure only includes the stiffness characteristics of the connecting structure, and the damping effect of the connecting substructure is not considered for the time being.

[0052] The left rotor substructure and connecting substructure are located in coordinate system O1X1Y1Z1, and the right rotor substructure is located in coordinate system O2X2Y2Z2. The two rotor parts rotate around axes Y1 and Y2 at speeds ω, respectively. Coordinate system O2X2Y2Z2 can be considered as a result of coordinate system O1X1Y1Z1 rotated by α around axis X1 and then translated to the right. Ignoring the rotor's rotational degrees of freedom around its axis, we can obtain the coordinate transformation matrix from coordinate system O1X1Y1Z1 to coordinate system O2X2Y2Z2 as shown in equation 1:

[0053] Where α is the angle between the two rotor shafts.

[0054] Step S102: Establish the motion differential equations of the left and right rotor substructures in their respective coordinate systems.

[0055] The generalized coordinates of the right rotor substructure are given by Equation 2: Where x i y i z i Let θ be the translational degree of freedom of the i-th node. xi and θ zi Let i be the rotational degrees of freedom of the i-th node, where i = 5, 6, 7, 8.

[0056] The right rotor substructure is located in coordinate system O2X2Y2Z2 and rotates about the Y2 axis at a rotational speed ω, resulting in the kinetic energy equation 3 for the right rotor substructure: Where m i =diag{m i m i mi J di J di}. Where, m i =diag{m i m i m i J di J di}, m i For the quality of the i-th node, J pi and J di Let be the polar moment of inertia and the cross-sectional moment of inertia of the i-th node, respectively.

[0057] Potential energy formula 4 for the right rotor substructure:

[0058]

[0059] in These are the stiffness matrices of the axis segments between nodes 5 and 6, 6 and 7, and 7 and 8 in the coordinate system O2X2Y2Z2, respectively. Let be the stiffness matrix of the bearings at nodes 6 and 8 in the coordinate system O2X2Y2Z2.

[0060] Substituting equations 3 and 4 into the Lagrange equations, we obtain equation 5, the differential equation of motion for the right-side rotor substructure:

[0061]

[0062] in in

[0063] in

[0064] Additional loads include misalignment loads caused by changes in the state of the connecting substructures.

[0065] In this embodiment, the method for establishing the motion differential equations of the left rotor substructure and the right rotor substructure in their respective coordinate systems is as follows:

[0066] The generalized coordinates of the left rotor substructure are given by Equation 6. Where x i y i z i Let θ be the translational degree of freedom of the i-th node. xi and θ zi Let i be the rotational degrees of freedom of the i-th node, where i = 1, 2, 3, 4.

[0067] Using the same method as for obtaining the differential equation of motion for the right rotor substructure, we obtain the differential equation of motion for the left rotor substructure. (Equation 7) in

[0068] Step S103: Perform coordinate transformation on the boundary nodes of the connecting substructure to establish the cross-coordinate system stiffness equation of the connecting substructure. There is an included angle between the axes on both sides of the connecting substructure, and the connecting structure has an initial deformation form 8. Connect the right node coordinates of the substructure and the left node The coordinate transformation, which takes place in different coordinate systems, yields the relative transformation form 9 connecting the two sides of the substructure: in Let I be the cross-coordinate system transformation matrix, and let I be the identity matrix. It is the set of mixed coordinate vectors of two boundary nodes;

[0069] Equation 10 for the potential energy of the connecting substructure in its own coordinate system k In the formula This is the stiffness matrix of the connecting substructure in coordinate system 1;

[0070] Substituting equation 9 into equation 10, we obtain equation 11:

[0071]

[0072] Substituting the potential energy of the connecting substructure into the Lagrange equation yields Equation 12.

[0073] in The cross-coordinate system stiffness matrix of the connecting substructure is constant because the two coordinate systems are fixed and time-varying issues are not involved during coordinate transformations. The right side represents the bending moment load caused by the initial angular offset of the connecting substructure. Since the direction of the angular misalignment does not change relative to the rotor system's foundation, the magnitude and direction of the resulting additional bending moment load remain unchanged.

[0074] Step S104: Based on the node positions of the connecting substructures, the motion differential equations of the left rotor substructure, the right rotor substructure, and the cross-coordinate system stiffness equations of the connecting substructures are assembled to form the entire cross-coordinate system motion differential equations of the rotor.

[0075] The connecting substructure contains two nodes, which belong to different coordinate systems O1X1Y1Z1 and O2X2Y2Z2. The cross-coordinate system stiffness matrix of the connecting substructure is divided into blocks according to the coordinate systems of the two nodes, as shown in Equation 13:

[0076]

[0077] Equation 13 can be rewritten as Equation 14:

[0078]

[0079] The stiffness matrix and load vector of the left and right rotors are divided into blocks, with J representing the degrees of freedom at the boundary nodes and I representing the degrees of freedom excluding the boundary nodes. Finally, the motion differential equations of the two rotors and the stiffness equations of the connecting substructure are assembled according to the corresponding nodes to obtain the complete cross-coordinate system motion differential equation 15 of the rotor:

[0080]

[0081] in:

[0082]

[0083]

[0084] As can be seen from the above modeling process and motion differential equations, the two rotor structures are modeled in coordinate systems with their respective axes Y1 and Y2. The model characteristics and motion differential equations are no different from those of traditional models. The only difference is that the two rotors are coupled in degrees of freedom by a cross-coordinate system stiffness matrix.

[0085] The principle of the cross-coordinate system dynamic modeling method for a rotor with angular misalignment fault provided by this invention is as follows: Angular misalignment faults in rotor structures generally occur at the connecting structures, where the rotor rotation axes on both sides of the connecting structure have a certain angle. Therefore, based on the axial characteristics of the rotor, the rotor model is divided into several substructures, each located in a different coordinate system. Each substructure is modeled in its own coordinate system, and the degrees of freedom of the entire rotor are coupled by transforming the coordinates of the connecting substructures. The degrees of freedom requiring coordinate transformation are limited to the connecting substructures; the degrees of freedom of the substructures on both sides of the connecting substructure are retained in physical space to facilitate degree-of-freedom reduction.

[0086] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0087] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this application are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of this application described herein can be implemented, for example, in sequences other than those illustrated or described herein.

[0088] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A cross-coordinate system dynamic modeling method for a rotor with misalignment fault, characterized in that, Includes the following steps: Based on the rotor's axial characteristics, the rotor experiencing misalignment is divided into a connecting substructure, a left rotor substructure, and a right rotor substructure. The left rotor substructure, connecting substructure, and right rotor substructure are located in different coordinate systems. The left rotor substructure includes nodes 1 to 4, and the right rotor substructure includes nodes 5 to 8. The mass of the connecting substructure is concentrated in nodes 4 and 5. The left rotor substructure and connecting substructure are located in coordinate system O1X1Y1Z1, and the right rotor substructure is located in coordinate system O2X2Y2Z2. The two rotor parts rotate around axes Y1 and Y2 at speeds ω, respectively. Coordinate system O2X2Y2Z2 is obtained by rotating coordinate system O1X1Y1Z1 by α around axis X1 and then translating it to the right. Ignoring the rotor's rotational freedom around the axis, the coordinate transformation matrix from coordinate system O1X1Y1Z1 to coordinate system O2X2Y2Z2 is given by equation 1: Where α is the angle between the two rotor shafts; Establish the motion differential equations of the left and right rotor substructures in their respective coordinate systems; Perform coordinate transformation on the boundary nodes of the connecting substructure to establish the cross-coordinate system stiffness equation of the connecting substructure; Based on the node positions of the connecting substructures, the motion differential equations of the left rotor substructure, the right rotor substructure, and the cross-coordinate system stiffness equations of the connecting substructures are combined to form the entire cross-coordinate system motion differential equations of the rotor.

2. The method for cross-coordinate system dynamic modeling of a rotor with misalignment fault according to claim 1, characterized in that, In the steps of establishing the differential equations of motion for the left and right rotor substructures in their respective coordinate systems, the method for establishing the differential equations of motion for the right rotor substructure is as follows: The generalized coordinates of the right rotor substructure are given by Equation 2: Where x i y i z i Let θ be the translational degree of freedom of the i-th node. xi and θ zi Let i be the rotational degrees of freedom of the i-th node, where i = 5, 6, 7, 8; The right rotor substructure is located in the coordinate system O2X2Y2Z2 and rotates about the Y2 axis at a rotational speed ω, resulting in the kinetic energy equation 3 for the right rotor substructure: in, m i =diag{m i m i m i J di J di }, m i For the quality of the i-th node, J pi and J di Let be the polar moment of inertia and the cross-sectional moment of inertia of the i-th node, respectively. Potential energy formula for the right rotor substructure: in and These are the stiffness matrices of the axis segments between nodes 5 and 6, 6 and 7, and 7 and 8 in the coordinate system O2X2Y2Z2, respectively. Let be the stiffness matrix of the bearings at nodes 6 and 8 in coordinate system O2X2Y2Z2; Substituting equations 3 and 4 into the Lagrange equations, we obtain equation 5, the differential equation of motion for the right-side rotor substructure: in in in Additional loads include misalignment loads caused by changes in the state of the connecting substructures.

3. The method for cross-coordinate system dynamic modeling of a rotor with misalignment fault according to claim 2, characterized in that, In the steps of establishing the differential equations of motion for the left and right rotor substructures in their respective coordinate systems, the method for establishing the differential equations of motion for the left rotor substructure is as follows: The generalized coordinates of the left rotor substructure are given by Equation 6: Where x i y i z i Let θ be the translational degree of freedom of the i-th node. xi and θ zi Let i be the rotational degrees of freedom of the i-th node, where i = 1, 2, 3, 4; Using the same method as for obtaining the differential equation of motion for the right rotor substructure, we obtain equation 7 of motion for the left rotor substructure: in 4. The method for cross-coordinate system dynamic modeling of a rotor with misalignment fault according to claim 3, characterized in that, The coordinate transformation of the boundary nodes of the connecting substructure and the establishment of the cross-coordinate system stiffness equation of the connecting substructure include: there is an angle between the axes on both sides of the connecting substructure, and the connecting structure has an initial deformation form 8: Connect the coordinates of the right node of the substructure and the left node The coordinate transformation, which takes place in different coordinate systems, yields the relative transformation form 9 connecting the two sides of the substructure: in Let I be the cross-coordinate system transformation matrix, and let I be the identity matrix. It is the set of mixed coordinate vectors of two boundary nodes; Equation 10 for the potential energy of the connecting substructure in its own coordinate system k In the formula This is the stiffness matrix of the connecting substructure in coordinate system 1; Substituting equation 9 into equation 10, we obtain equation 11: Substituting the potential energy of the connecting substructure into the Lagrange equation yields Equation 12. in This is the cross-coordinate system stiffness matrix connecting the substructures.

5. The cross-coordinate system dynamic modeling method for a rotor with misalignment fault according to claim 4, characterized in that, Based on the node positions of the connecting substructures, the motion differential equations of the left and right rotor substructures, as well as the cross-coordinate system stiffness equations of the connecting substructures, are assembled to form the entire rotor cross-coordinate system motion differential equations, including: The cross-coordinate system stiffness matrix of the connecting substructure is divided into blocks according to the coordinate systems of the two nodes, as shown in Equation 13: Equation 12 can be rewritten as Equation 14. The stiffness matrix and load vector of the left and right rotors are divided into blocks, with J representing the degrees of freedom at the boundary nodes and I representing the degrees of freedom excluding the boundary nodes. Finally, the motion differential equations 5 and 7 of the two rotors and the stiffness equation 14 of the connecting substructure are assembled according to the corresponding nodes to obtain the complete cross-coordinate system motion differential equation 15 of the rotor. in:

Images