Rotor centering fault dynamic model for gear coupling of rotary machinery

Abstract

According to the rotor centering fault dynamic model of the rotary mechanical gear coupling, the motion state and the stress condition of the coupling under the non-centering condition are analyzed, the radial force change caused by different torque transmission in the gear meshing process is obtained, and the non-centering force exciting force expression applied to a rotor system by the gear coupling is obtained; establishing a dynamic vibration equation of the rotor system, solving to obtain an overall unit stiffness matrix, a mass matrix and a damping matrix of the shaft system, and deducing a motion differential equation of the rotor system; the method comprises the following steps of: obtaining a displacement value, a speed value and an acceleration value of each node at each moment, drawing an axis trajectory diagram, a time domain oscillogram, a spectrogram and a poincare sectional view to analyze the dynamic behavior of a coupling on a transit subsystem and the action of the coupling on the motion characteristics of the coupling by parameters of a non-centering quantity, rotating speed and torque system, and analyzing the stability of the system under different rotating speeds to obtain the stability of the system. And great help can be provided for preventing, analyzing and processing the centering fault of the coupler rotor.

Description

Technical Field

[0001] The present application relates to a misalignment fault model of a rotating coupling rotor, and particularly to a dynamic model of a misalignment fault of a toothed coupling rotor of a rotating machine, belonging to the technical field of misalignment faults of coupling rotors. Background Technique

[0002] Rotating machines are widely used in generators, compressors, steam turbines, etc. in electric power, petrochemical industry, and transportation, and are extremely important components in the industrial production process, which is related to the safe and stable operation of the entire system. A rotating machine is a machine with a rotor as the core, and its dynamic characteristics have a decisive impact on the safe and stable operation and working performance of the rotating machine. Among the causes of rotating machine failures, the misalignment fault of the rotor system accounts for more than 60%. Because the toothed coupling has the characteristics of a small radial dimension, the ability to bear a large load, and the ability to adjust the rotor misalignment, it is widely used. Therefore, it is necessary to study the motion mechanism of the toothed coupling and the causes of the misalignment force, as well as the dynamic behavior of the rotor system when there is a misalignment in the toothed coupling. This is of great significance for early detection of faults and prevention of the resulting adverse vibrations and potential safety hazards to the system.

[0003] During the processing, installation, and operation of rotating machines, due to various reasons such as poor machining accuracy, manual installation errors, and different material thermal deformation coefficients during operation, the occurrence of misalignment faults in the system will ultimately be triggered. When the center lines of the two rotating shafts are not collinear with the center lines of the bearings, the rotor system is misaligned. This will not only cause alternating stresses in the axial and radial directions of the system, resulting in vibrations in both directions, but also cause friction damage to the bearings under the action of additional stresses, and at the same time cause collisions and wear between the rotor and the stator, as well as problems such as deflection deformation of the rotating shaft, vibration, and noise, endangering the stable operation of the system.

[0004] In a rotor system, if there is a misalignment fault form, there are two cases: bearing misalignment and coupling misalignment. If the journal has a deflection angle or elevation change in the bearing, the bearing will have a misalignment fault, which will form an additional bending moment at the coupling position, and the influence on the system is reflected in the oil film characteristics and damping. When there is an imbalance fault at the same time, there will be a reaction force on the imbalance, causing the system to generate power frequency vibration. Coupling misalignment means that the axes of the two half-couplings are not collinear, or it is not a very continuous and smooth straight line, that is, there are step points or inflection points at the position of the half-coupling. According to the states of the two half-couplings, the misalignment can be divided into: parallel misalignment, angular misalignment, and comprehensive misalignment that covers both parallel and angular misalignment types. When there is a misalignment fault in the coupling, an additional bending moment and shear force will be formed at the position where the two rotating shafts are connected. At the same time, the radial force of the bearing close to it will increase, making the stress condition of the bearing deteriorate, resulting in the generation of system vibration and having an adverse impact on the rotor system.

[0005] The problems to be solved in the prior art for the alignment fault diagnosis of the toothed coupling rotor and the key technical difficulties of this application include:

[0006] (1) In the prior art, during the machining and installation process of rotating machinery, due to reasons such as poor machining accuracy, manual errors in installation, and different thermal deformation coefficients of materials during operation, the alignment fault of the system is ultimately caused. This will not only cause the system to generate alternating stresses in the axial and radial directions, resulting in vibrations in both directions, but also cause frictional damage to the bearing under the action of additional stresses. At the same time, it will cause collisions and wear between the rotor and the stator, as well as flexural deformation of the rotating shaft, generating vibrations and noises, etc., endangering the stable operation of the system. The prior art lacks an effective and reliable method for diagnosing the alignment fault of the toothed coupling rotor, does not establish a motion force analysis model for the coupling, lacks the analysis of the motion force law of the coupling, and thus also lacks a fault model for the alignment of the toothed coupling in the rotor system. It is impossible to analyze the motion mechanism of the toothed coupling and the cause of the misalignment force, as well as the dynamic behavior of the rotor system when there is a misalignment amount in the toothed coupling, which is not conducive to early detection of faults and prevention of the resulting adverse vibrations and potential safety hazards to the system.

[0007] (2) The existing technologies lack a dynamic model for the misalignment fault of a gear coupling rotor. Starting from the three fault states of the misaligned gear coupling, the force analysis and motion research of the coupling under each fault condition have not been carried out, the motion laws of the two half-couplings and their tooth sleeves in different situations cannot be deduced, the expression formula of the radial force generated during the meshing transmission of the coupling cannot be obtained, and the variation law of the radial force at the position of each tooth during the internal and external tooth meshing of the gear coupling with system parameters is lacking. The existing technologies have not established the motion differential equation of the rotor system, have not considered the influence of oil film force, unbalance force and gravity, cannot discretize the rotor system model, cannot obtain the overall mass matrix, overall stiffness matrix and overall damping matrix of the system, have not solved the response of the vibration equation of the faulty rotor system, cannot obtain the displacement values, velocity values and acceleration values at each node position of the misaligned rotor system of the gear coupling, and cannot study the vibration response characteristics of the key parts of the rotor system with changes in system parameters such as rotational speed and misalignment amount and the stability situation with changes in rotational speed, which is not conducive to the prevention, analysis and treatment of the misalignment fault of the coupling rotor.

[0008] (3) The current research on the misalignment problem of couplings focuses on the system response characteristics and acting mechanisms when the coupling misalignment fault occurs, the coupling fault forms and their vibration laws of complex rotor systems, etc. There is a lack of discovering and eliminating problems through the motion behavior and state of the rotor system before the misalignment fault occurs, which is not conducive to reducing the probability of fault occurrence and avoiding unnecessary losses. There is a lack of analysis of the three types of misalignment faults existing in the gear coupling, a lack of motion forms and force conditions under different faults, the motion laws of the half-couplings and the coupling tooth sleeves cannot be obtained, and there is a lack of simulation analysis of the radial force generated during the internal and external tooth meshing of the coupling. The mathematical model of the rotor system when the gear coupling has a misalignment fault has not been established, the expression forms of the system excitation forces and the system matrix have not been deduced, and the methods and steps for solving the motion differential equation of the system are lacking. There is a lack of simulation of the vibration characteristics of the rotor system when the gear coupling has a misalignment fault, and there is a lack of analyzing the vibration response and stability situation of the key node positions on the coupling node and the shafting with changes in system parameters through waveforms and spectra. The accuracy and timeliness of fault identification and analysis are relatively poor. Summary of the Invention

[0009] This application analyzes the motion state and force condition of the misaligned coupling, obtains the radial force change caused by different torque transmissions during gear meshing, and simultaneously considers the excitation force applied to the system due to the centrifugal effect during the movement of the coupling bushing, obtains the misalignment force excitation force expression applied by the toothed coupling to the rotor system, and establishes a mathematical model of the misalignment force when the coupling has a misalignment fault; based on the action of oil film force, unbalance force, and gravity, establishes the dynamic vibration equation of the rotor system, discretizes the rotor system, solves to obtain the overall element stiffness matrix, mass matrix, and damping matrix of the shafting, and derives the motion differential equation of the rotor system; solves the differential equation of the rotor system, obtains the displacement values, velocity values, and acceleration values of each node at each moment, and draws the axis locus diagram, time-domain waveform diagram, frequency spectrum diagram, and Poincare section diagram to analyze the dynamic behavior of the rotor system with coupling misalignment, as well as the effects of system parameters such as misalignment amount, rotational speed, and torque on its motion characteristics, and analyzes the stability of the system at different rotational speeds, obtaining a powerful dynamic model of the toothed coupling rotor misalignment fault, which can be of great help in preventing, analyzing, and dealing with the misalignment fault of the coupling rotor.

[0010] To achieve the above technical effects, the technical solutions adopted in this application are as follows:

[0011] A dynamic model of the toothed coupling rotor misalignment fault in rotating machinery, which analyzes the motion state and force condition of the misaligned coupling, obtains the radial force change caused by different torque transmissions during gear meshing, and simultaneously considers the excitation force applied to the system due to the centrifugal effect during the movement of the coupling bushing, obtains the misalignment force excitation force expression applied by the toothed coupling to the rotor system, and establishes a mathematical model of the misalignment force when the coupling has a misalignment fault; based on the action of oil film force, unbalance force, and gravity, establishes the dynamic vibration equation of the rotor system, discretizes the rotor system, solves to obtain the overall element stiffness matrix, mass matrix, and damping matrix of the shafting, and derives the motion differential equation of the rotor system; solves the differential equation of the rotor system, obtains the displacement values, velocity values, and acceleration values of each node at each moment, and draws the axis locus diagram, time-domain waveform diagram, frequency spectrum diagram, and Poincare section diagram to analyze the dynamic behavior of the rotor system with coupling misalignment, as well as the effects of system parameters such as misalignment amount, rotational speed, and torque on its motion characteristics, and analyzes the stability of the system at different rotational speeds. The rules include:

[0012] 1) At the same meshing point, the circumferential force is greater than the radial force; as the misalignment amount increases, the maximum value of the radial force increases, and the minimum value decreases relatively. The greater the misalignment amount, the greater the change rate of the radial force near the meshing point at the upper end of the coupling, that is, the faster the meshing force of the coupling changes; when the transmitted torque value is larger, the maximum value of the radial force borne by the coupling is larger, and the minimum value is also relatively larger. The change rate of the radial force near the upper end of the coupling is relatively small, that is, the impact and vibration caused by the change of the coupling misalignment amount on the system are greater than the influence when the torque changes;

[0013] 2) When the rotor system is well-aligned, the axis locus is elliptical, the time-domain waveform is a sine wave curve, and there is only a 1x frequency component on the frequency spectrum diagram, and the system is affected by the unbalanced force; when the toothed coupling is misaligned, the axis locus is banana-shaped or inner 8-shaped, and it is banana-shaped when the misalignment amount is small (the rotational speed is large), and inner 8-shaped when the misalignment amount is large (the rotational speed is small), that is, its deformation degree is affected by the misalignment amount and the rotational speed. The time-domain waveform diagram is a double-peak waveform curve, and a 2x frequency component appears on the frequency spectrum diagram in addition to the 1x frequency; the misalignment amount and torque affect the magnitude of the misalignment excitation force of the toothed coupling, and the rotational speed not only affects the magnitude of the misalignment force and the unbalanced force, but also changes the distribution direction of the misalignment force in the system;

[0014] 3) The vibration response at different positions of the shafting changes with the system parameters: the amplitudes of the 1x and 2x frequency components are larger at the position closer to the coupling. When the toothed coupling is misaligned, the vibration near it is more obvious; as the misalignment offset increases, the amplitude of the 1x frequency component remains unchanged, and the amplitude of the 2x frequency component increases continuously. The misalignment amount affects the magnitude of the misalignment force and has no effect on the unbalanced force; the amplitude of the 1x frequency component on the left side of the shafting is smaller, and the amplitude of the 2x frequency component is larger. The unbalanced force is relatively smaller and the misalignment force is relatively larger on the side with a smaller shaft mass.

[0015] Preferably, the motion state analysis of the toothed coupling: When the axes of the two half-couplings are parallel to the design axis or have a certain angle with it, it is considered that the coupling has an alignment fault. If the coupling has an alignment fault, there are parallel misalignment, angular misalignment, and mixed misalignment faults including parallel and angular misalignment; magnify the locus diagram of the rotation center part, where o 1 represents the rotation center of the left half-coupling, o 2 represents the rotation center of the right half-coupling, and o' represents the center of the coupling sleeve in the motion state. Under the alignment fault, o' is a circle with o as the center and △y as the radius. In the isosceles triangle △o 1 oc, from the geometric relationship, we get: ω c t = 2∠o′o 1 o = 2ωt, that is, ω c = 2ω, and the rotation frequency of the coupling sleeve is twice the system frequency;

[0016] Let X and Y be the moving coordinates of o' in the coordinate system respectively, and its expression is obtained as follows:

[0017]

[0018] Similarly, the motion trajectory of the instantaneous center c' is:

[0019]

[0020] The motion equation of the coupling meshing circle is:

[0021] (X - Δy sin 2ωt) 2 +(Y - Δy cos 2ωt) 2 =R i 2 Equation 3

[0022] When R i Correspondingly, when taking R 1 、R 2 、R, they respectively represent the trajectories formed by the inner meshing circle, the outer meshing circle and the pitch circle during the motion of the toothed coupling.

[0023] Preferably, analyze the radial force generated by misalignment: During the meshing of gears, the normal force F acting on a single tooth surface n is decomposed into a circumferential force and a radial force perpendicular to each other. Among them, the circumferential force F t is tangent to the meshing circle of the gear, and the radial force F r points vertically downward to the axis, and their magnitudes are respectively:

[0024]

[0025] Let T be the magnitude of the torque transmitted by the coupling in the rotor system, R be the radius of the gear meshing circle, and α be the pressure angle of the meshing circle. When a toothed coupling misalignment fault occurs in the rotor system, based on the misalignment offset, the inner and outer gears are most closely meshed at the upper end point of the coupling. Let the torque transmitted at this point be dT and the meshing angle be α 1 , and the normal force dF on the tooth surface at this meshing point is obtained n as:

[0026]

[0027] The magnitude of the radial force on the tooth surface is:

[0028]

[0029] Similarly, let the meshing angle at the meshing point 2 be α 2 , then the magnitude of its radial force on the tooth surface is:

[0030]

[0031] dT 1 Much larger than dT 2 , then dF r1 >>dF r2 , the radial force magnitudes at other meshing points are between dF r1 and dF r2 . In the case of good alignment of the coupling, the motion centers of the left and right half-couplings coincide with the motion center of the coupling sleeve at one point. Let the distances between the root circles and the meshing circles of the left and right half-couplings be c at this time. The radial force magnitudes at each meshing point are equal and point from the center of the circle to the coupling sleeve; when the misalignment offset of the coupling is △y, the minimum clearance between the coupling sleeve and the half-coupling is at the upper end of the gear, and its magnitude is: C min = c - △y, the maximum clearance at the lower end is C max = c + △y, the tooth clearance near the left side of the upper end is C 2 = c - △ycosθ 1 , the tooth clearance near the right side of the lower end is C 2 ’ = c + △ycosθ 1 . By analogy, the tooth clearance expressions for each tooth are obtained, and their magnitudes are related to the misalignment amount △y, the tooth clearance c under alignment, and the angle θ i of each tooth.

[0032] Let the number of teeth of the gear coupling be z. Starting from the tooth at the uppermost end, it is sequentially numbered 1, 2, 3... z / 2 + 1, z / 2 + 2…z in the counterclockwise direction, and the angle θ i of each tooth is: θ i = 2πi / z. Each tooth is regarded as a cantilever beam. The position of the axis at rest is y, and the position of the deformed axis during the meshing process is y'. The deflection magnitudes at each meshing point are equal, that is, it satisfies v 1 = v 2 = … = v i = … = v z . Therefore, at the 1st tooth, we get:

[0033] M 1 (y) = F t1 [(C - Δe) - y] Equation 8

[0034] EIv″ = F t1 [(c - Δe) - y] Equation 9

[0035] where M 1 is the bending moment magnitude at the meshing point position of the 1st tooth, and F t1is the circumferential force magnitude of the No. 1 tooth at the meshing point. EI represents the flexural rigidity of the No. 1 tooth. v” is the second derivative value of the deflection at the position of the No. 1 tooth. Integrating it gives:

[0036] According to the boundary conditions of the fixed end, we get:

[0037]

[0038]

[0039] According to the boundary conditions of the fixed end, we get:

[0040]

[0041] Then we have:

[0042] The deflection of the gear coupling at the No. 1 tooth is:

[0043]

[0044] The deflection at the meshing point of the No. 2 tooth is:

[0045]

[0046] According to the deformation compatibility condition, the deflections of each tooth are the same. Then:

[0047]

[0048] It is deduced that:

[0049] Without considering the change in the lever arm caused by the misalignment of the gear coupling, it is considered to be always equal to d / 2. Then the total torque is the sum of the torques transmitted by each tooth, and its expression is:

[0050]

[0051] Let:

[0052]

[0053] Then we have: F t1 = 2T / dgE. Substituting it in, the tangential force received by the Nth tooth is:

[0054]

[0055] Then the magnitude of the radial force received by the Nth tooth of the coupling is:

[0056]

[0057] Substitute F rNDecompose along the x-axis to obtain the radial force formed due to unequal torque transmission, and its magnitude is:

[0058]

[0059] Its direction is from the point where the combination is the tightest (the meshing point of tooth No. 1) to the point where the combination is the loosest (the meshing point of tooth z / 2 + 1) during the gear meshing process.

[0060] Preferably, the law of force on the coupling during movement: Conduct a dynamic analysis on the forms and motion states of the alignment faults of the toothed coupling, obtain the motion trajectories and characteristics of the left and right half couplings and the coupling sleeve under different conditions, establish a radial force model generated during the coupling meshing process, and obtain its distribution at each tooth and the variation law under different misalignments and torques, including:

[0061] 1) When the misaligned toothed coupling is in a normal motion state, the frequency of the coupling sleeve is equal to twice the rotational frequency of the rotor; when the toothed coupling fails to meet the rotation condition and is locked, the motion frequency of its sleeve is the same as the frequency of the rotor system;

[0062] 2) When there is a parallel misalignment fault state in the toothed coupling, the phase difference between the end faces perpendicular to the axis of the left and right half couplings is 0 degrees, while in the angular misalignment fault state, the phase difference between the two is 180 degrees, and in the comprehensive misalignment fault state, the phase difference is between the two. Based on the known 2x frequency characteristics, distinguish the types of alignment faults existing in the coupling in the rotor system;

[0063] 3) When the coupling is in a parallel alignment fault state, the motion trajectory of the axis of the coupling sleeve is a cylinder, while when it is in an angular alignment fault state, the motion trajectory is a double-cone curve, and the motion range is related to the misalignment offset. In the comprehensive alignment fault state, the motion shape of the axis of the coupling sleeve is between a cylinder and a double-cone, which is a semi-double-cone.

[0064] Preferably, the alignment fault model of the toothed coupling in the rotor system: The rotor system includes discrete impellers, shaft segments, and supports. Discretize the system along the axis direction, and simplify it into a model composed of rigid discs, shaft segments, and bearing seats connected by nodes. The center of the disc, the center of the journal, the center of the bearing seat, and a certain point on the axis can all be selected as nodes and numbered in sequence, then the discretized rotor system model is obtained. Establish the motion equation of the rotor system when there is an alignment fault state in the toothed coupling, as shown in Equation 22:

[0065]

[0066] Among them, [M] is the overall mass matrix of the system, including the rotational mass matrix and the translational inertia matrix; [C] represents the overall damping matrix of the system, including the damping matrix of the system and the gyroscopic force matrix; [K] represents the overall stiffness matrix of the rotor system; z, respectively represent the displacement value, velocity value, and acceleration value at the position of the node after the system is discretized; F r (z, t) represents the misalignment excitation force matrix when there is a misalignment offset in the gear coupling; is the non-linear oil film force array, Q(t) is the unbalance force array of the system; G is the gravity of the system, Q(t), F r (z, t), and the sum of G is the excitation force array received by the rotor system of the gear coupling in the misalignment fault state.

[0067] Preferably, the misalignment coupling meshing radial excitation force model: o 1 、o 2 are respectively the rotation centers of the left and right half couplings. During the meshing process, due to the difference in the torque transmitted by each tooth between the left and right half couplings, the magnitudes of the radial forces generated are F r合 、F R合 , o 1 、o 2 are not at the same point, and F r合 points from o 1 to o', F R合 points from o 2 to o';

[0068] At a certain moment when the system is in misaligned motion, assume that the coordinates of the rotation center o 1 of the left half coupling are (x i , y i ), and the coordinates of the rotation center o 2 of the right half coupling are (x i+1 , y i+1 ), then there is:

[0069]

[0070] e 1 、e 2 respectively represent the distances from the rotation centers o 1 、o 2 of the left and right half couplings to the rotation center o' of the coupling sleeve during movement, then there is:

[0071] e 1 =o 1 o′=Δysinωt Equation 24

[0072] e 2= o 2 o' = Δycosωt Equation 25

[0073] The radial force F of the left half coupling r合 The components on the x and y axes are respectively:

[0074] F rx = -F r合 (e 1 )cosωt Equation 26

[0075] F ry = -F r合 (e 1 )sinωt Equation 27

[0076] For the components of the radial force F of the right half coupling on the x and y axes, there are:

[0077]

[0078] The meshing radial excitation force model of the misaligned coupling is obtained.

[0079] Preferably, the centrifugal excitation force model of the misaligned coupling tooth sleeve: In the misalignment fault state, the rotation frequency of the coupling tooth sleeve is 2 times the frequency of the rotor system. Then, the components f rx 、f ry of the centrifugal force exerted by the tooth sleeve movement on the left half coupling on the x and y axes are respectively:

[0080] f rx = -m c e 1 (2ω) 2 cos(ωt) Equation 30

[0081] f ry = -m c e 1 (2ω)2sin(ωt) Equation 31

[0082] The components f Rx 、f Ry of the centrifugal force exerted by the right half coupling under the action of the coupling tooth sleeve on the x and y axes are:

[0083] f Rx = -m c e 2 (2ω) 2 cos(ωt) Equation 32

[0084] f Ry = -m c e 2 (2ω) 2 sin(ωt) Equation 33

[0085] The misalignment excitation force on the rotor system is the sum of the radial force generated by torsion and the centrifugal force generated by the eccentricity of the gear sleeve, that is, the magnitudes of the misalignment forces on the left and right half couplings are as follows:

[0086] F xi = F rx + f rx

[0087] F yi = F ry + f ry Equation 34

[0088] F x(i+1) = F Rx + f Rx

[0089] F y(k+1) = F Ry + F Ry Equation 35

[0090] Where i and i + 1 represent the positions of the left and right half couplings in the discrete system, then:

[0091]

[0092] The above expression form of the misalignment excitation force in the motion differential equation of the rotor system.

[0093] Preferably, the rotor unbalance excitation force model: The mass eccentricity of the disk in the rotor system will apply an unbalance excitation force to the system during motion. Let its initial eccentricity phase be The magnitude of the disk eccentricity is e, and the mass of the disk is m p , when the rotational speed of the rotor is ω, the components Q x , Q y of the unbalance force on the x and y axes are respectively:

[0094]

[0095]

[0096] The expression form of the unbalance excitation force in the system vibration equation is as follows:

[0097]

[0098] Where i represents the position of the disk in the discrete system.

[0099] Preferably, the matrix model of the rotor system: The misaligned rotor system of the gear coupling is composed of discrete elastic shaft segment units, and its generalized coordinates are the displacements at both ends of the shaft segment. Ignoring its axial deformation, its generalized coordinates are:

[0100]

[0101] where x and y are the displacements at the axis center position, and θ x , θ y are the deflection angles of the cross-section, both of which are functions of the position s and time t. The displacement at any cross-section is transformed using the displacement difference function [N], [N] = [N 1 (s), N 2 (s), N 3 (s), N 4 (s)]. Then the expression for the corresponding displacement is:

[0102] x(s, t) = [N]{u 1s} Equation 41

[0103] where [N} = [N 1 (s), N 2 (s), N 3 (s), N 4 (s)]. Then the expression for the corresponding displacement is:

[0104]

[0105] According to the boundary conditions at the endpoints, we get:

[0106]

[0107] Then the constraints of the interpolation function are:

[0108]

[0109] Assume the above interpolation function to be in the form of a cubic polynomial with respect to the position s:

[0110] N 1 (s) = a 0 + a 1 s + a 2 s 2 + a 3 s 3 Equation 45

[0111] Similarly, we get:

[0112] y(s, t) = [N]{u 2s} Equation 46

[0113] The displacements at the positions of each cross-section in this unit are:

[0114]

[0115] The mass per unit length of the shaft segment unit is u, and its polar moment of inertia is j v , and the magnitude of the diametral moment of inertia is j d , for an infinitesimal element with an axial distance of s from node A and a thickness of ds, the expression for its bending elastic potential energy is obtained:

[0116]

[0117] Integrating it step by step along the shaft segment unit with a radius of r and a length of 1, the kinetic energy and potential energy of the shaft segment can be obtained as:

[0118]

[0119] Obtained: J ps = j ps 1, where:

[0120]

[0121] [M sT is the translational inertia matrix of the element, [M sR is the rotational inertia matrix of the element, [K s is the stiffness mass matrix of the element, where the gyroscopic force matrix is [G s J = ω[J s , the mass matrix [M s of the element is a consistent mass matrix considering the rotational inertia and translational inertia of the shaft segment, that is: [M s = [M sT + [M sR ;

[0122] When the discrete rotor system is composed of N nodes and N - 1 shaft segments connected, its overall mass matrix [M], overall stiffness matrix [K], and overall gyroscopic force matrix [G] can all be obtained by superimposing the element mass matrix, unit stiffness matrix, and unit gyroscopic matrix, and they are all matrices of order 4N×4N.

[0123] Preferably, the numerical solution method of the rotor system motion equation: Obtain the displacement values, velocities, and acceleration values generated at the node positions under the action of each excitation force of the faulty rotor system, and the solution steps are as follows:

[0124] X’ t+Δt = X’ t + (1 - γ)Δtx” t + γΔtx’ t+Δt Equation 52

[0125]

[0126] Among them, γ and β are parameters adjusted according to the requirements of integration accuracy and stability. If γ = 0.5 and β = 1 / 6, the above formula is the linear acceleration method. Assuming 0 ≤ τ ≤ △t, the integral of the acceleration expression under the linear assumption within the time interval △t is obtained:

[0127] (1) Calculate the initial values

[0128] ① The overall mass matrix [M], overall stiffness matrix [K], and overall damping matrix [C] of the system are obtained by superimposing the element mass matrix, element stiffness matrix, element gyroscopic force matrix, and element damping matrix of the shaft segment;

[0129] ② Given the initial displacement x, initial velocity x', and initial acceleration x'' values of the rotor system;

[0130] ③ Determine the calculation time step △t and parameters γ, β, and obtain the integration constants:

[0131] ④ Obtain the effective stiffness matrix:

[0132] ⑤ Decompose the matrix

[0133] (2) Solve for each t + △t: the equivalent excitation force, displacement, velocity, and acceleration.

[0134] Compared with the prior art, the innovation points and advantages of this application are as follows:

[0135] (1) This application analyzes the motion state and force conditions of the misalignment of the coupling, obtains the change in the radial force caused by different torque transmissions during the gear meshing process, and at the same time considers the excitation force exerted on the system due to the centrifugal effect during the movement of the coupling bushing, obtains the expression of the misalignment force excitation force exerted by the toothed coupling on the rotor system, and establishes a mathematical model of the misalignment force when there is a misalignment fault in the coupling; based on the action of the oil film force, unbalance force, and gravity, establish the dynamic vibration equation of the rotor system, discretize the rotor system, solve to obtain the overall element stiffness matrix, mass matrix, and damping matrix of the shafting, and deduce the motion differential equation of the rotor system; solve the differential equation of the rotor system to obtain the displacement values, velocity values, and acceleration values of each node at each moment, draw the center orbit diagram, time-domain waveform diagram, frequency spectrum diagram, and Poincare section diagram to analyze the dynamic behavior of the rotor system with coupling misalignment, as well as the effects of misalignment amount, rotational speed, torque system parameters on its motion characteristics, and analyze the stability of the system at different rotational speeds, obtain a powerful dynamic model of the toothed coupling rotor misalignment fault, which can be of great help in preventing, analyzing, and dealing with the coupling rotor misalignment fault.

[0136] (2) The dynamic fault analysis of the coupling rotor alignment in this application shows that at the same meshing point, the circumferential force is greater than the radial force; as the misalignment amount increases, the maximum value of the radial force increases, and the minimum value decreases relatively. Moreover, the greater the misalignment amount, the greater the change rate of the radial force near the meshing point at the upper end of the coupling, that is, the faster the meshing force of the coupling changes; when the transmitted torque value is larger, the maximum value of the radial force borne by the coupling is larger, and the minimum value is also relatively larger. The change rate of the radial force near the upper end of the coupling is relatively small, that is, the impact and vibration caused by the change of the misalignment amount of the coupling on the system are greater than the influence when the torque changes; when the rotor system is well-aligned, the axis locus is elliptical, the time-domain waveform is a sine wave curve, and there is only a fundamental frequency component on the frequency spectrum diagram, and the system is affected by the unbalanced force; when the toothed coupling is misaligned, the axis locus is banana-shaped or inner 8-shaped, and it is banana-shaped when the misalignment amount is small (the rotational speed is large), and inner 8-shaped when the misalignment amount is large (the rotational speed is small), that is, its deformation degree is affected by the misalignment amount and the rotational speed. The time-domain waveform diagram is a double-peak waveform curve, and there is a second harmonic component in addition to the fundamental frequency on the frequency spectrum diagram; the misalignment amount and torque affect the magnitude of the misalignment excitation force of the toothed coupling, and the rotational speed not only affects the magnitude of the misalignment force and the unbalanced force, but also changes the distribution direction of the misalignment force in the system; it well analyzes the motion mechanism of the toothed coupling and the cause of the misalignment force generation, as well as the dynamic behavior of the rotor system when there is a misalignment amount in the toothed coupling, which is beneficial to early detection of faults and prevention of the resulting adverse vibrations and potential safety hazards to the system.

[0137] (3) This application obtains the vibration response at different positions of the shafting with the change of system parameters: the amplitudes of the fundamental frequency and second harmonic components are larger at the position closer to the coupling. When the toothed coupling is misaligned, the vibration near it is more obvious; as the misalignment offset increases, the amplitude of the fundamental frequency component remains unchanged, and the amplitude of the second harmonic component increases continuously. The misalignment amount affects the magnitude of the misalignment force and has no effect on the unbalanced force; the amplitude of the fundamental frequency component on the left side of the shafting is smaller, and the amplitude of the second harmonic component is larger. The unbalanced force is relatively small on the side with a smaller mass of the rotating shaft, and the misalignment force is relatively large. The displacement values, velocity values, and acceleration values at each node position of the toothed coupling misaligned rotor system are obtained, and through methods such as waveform analysis and time-domain analysis, the vibration response characteristics of the key parts of the rotor system with the change of system parameters such as rotational speed and misalignment amount and the stability situation with the change of rotational speed are analyzed. The accuracy and timeliness of the identification and analysis of the coupling rotor alignment fault have been greatly improved. Description of the Drawings

[0138] Figure 1 It is a comparison diagram of transformer fault types and characteristic gases.

[0139] Figure 2It is a schematic diagram of the form when the toothed coupling is in good alignment.

[0140] Figure 3 It is a schematic diagram of the form when the toothed coupling is in poor alignment.

[0141] Figure 4 It is a diagram of the force deformation of the teeth of the toothed coupling.

[0142] Figure 5 It is a schematic diagram of the meshing state relationship of the misaligned coupling.

[0143] Figure 6 It is a diagram of the positional relationship of each part of the toothed coupling.

[0144] Figure 7 It is a schematic diagram of the discretized beam element model of the rotating shaft. Specific implementation manner

[0145] The following combines the accompanying drawings to further describe the technical solution of the dynamic model of the toothed coupling rotor alignment fault provided by this application, so that those skilled in the art can better understand this application and be able to implement it.

[0146] This application first analyzes the motion state and force conditions under misalignment of the coupling, obtains the change in the radial force caused by different torque transmissions during the gear meshing process, and at the same time considers the excitation force applied to the system due to the centrifugal action during the movement of the coupling bushing, obtains the expression of the misalignment force excitation force applied by the toothed coupling to the rotor system, and establishes a mathematical model of the misalignment force when there is an alignment fault in the coupling; based on the action of the oil film force, unbalanced force, and gravity, establish the dynamic vibration equation of the rotor system, perform discretization processing on the rotor system, solve to obtain the overall element stiffness matrix, mass matrix, and damping matrix of the shafting, and derive the motion differential equation of the rotor system; solve the differential equation of the rotor system to obtain the displacement values, velocity values, and acceleration values of each node at each moment, and analyze the dynamic behavior of the rotor system with misaligned coupling by drawing the axis locus diagram, time-domain waveform diagram, frequency spectrum diagram, and Poincare section diagram, as well as the effects of misalignment amount, rotational speed, torque system parameters on its motion characteristics, and analyze the stability of the system at different rotational speeds.

[0147] I. Coupling motion force analysis model

[0148] (1) Analysis of the motion form of the toothed coupling

[0149] It is determined that if the relative positions of the left and right half couplings are completely in the initial design state, the alignment is good. When the axes of the two half couplings are parallel to the design axis or have a certain included angle with it, it is determined that the coupling has an alignment fault. If the coupling has an alignment fault, there are parallel misalignments, angular misalignments, and mixed misalignments that include both parallel and angular misalignments;

[0150] The trajectory diagram of the enlarged center of rotation part is as Figure 1 shown, where o 1 represents the center of rotation of the left half coupling, o 2 represents the center of rotation of the right half coupling, o' represents the center of the coupling sleeve in the motion state. Under the alignment fault, o' is a circle with o as the center and △y as the radius. In the isosceles triangle △o 1 oc, from the geometric relationship, we get: ω c t = 2∠o′o 1 o = 2ωt, that is, ω c = 2ω. The rotation frequency of the coupling sleeve is twice the system frequency;

[0151] Let X and Y be the motion coordinates of o' in the coordinate system respectively, and its expression is obtained as:

[0152]

[0153] Similarly, the motion trajectory of its instantaneous center c' is:

[0154]

[0155] The motion equation of the coupling meshing circle is:

[0156] (X - Δysin2ωt) 2 +(Y - Δycs2ωt) 2 = R i 2 Equation 3

[0157] When R i Correspondingly, when taking R 1 、R 2 、R, they respectively represent the trajectories formed by the inner meshing circle, outer meshing circle, and pitch circle during the motion of the toothed coupling;

[0158] In the state of parallel misalignment fault, the trajectory formed by the coupling sleeve along with the meshing motion of the left and right half couplings is: x 2 +y 2 = △y 2, is a cylinder. The coupling sleeve satisfies the misaligned motion state of the coupling. The center of mass o' moves around the static axis with an angular velocity of 2ω, and the phases of the two coupling end faces are the same. In the case of angular misalignment fault, the motion trajectory of its axis is: x 2 +y 2 = [(△L / 2 - z)tanα], which is a double-cone curve. And the rotation center o' of the coupling sleeve also moves around o at twice the system frequency, and the phases of the two half-coupling end faces differ by 180 degrees. In the case of comprehensive misalignment fault, the trajectory equation of the coupling sleeve axis is: x 2 +y 2 = [(ΔL / 2 - z)tanα + Δy / 2] 2 , is a double-cone. The center of mass o' rotates at a frequency of 2ω, with a 2x frequency characteristic, and the phase difference between the left and right half-coupling end faces is 0° to 180°.

[0159] (2) Analysis of the radial force generated by misalignment

[0160] Assume that the coupling sleeve of the toothed coupling is in a meshing state with straight teeth with the half-couplings. The pressure angle position during the meshing process of the coupling sleeve of the toothed coupling with the left and right half-couplings is fixed. Without considering the frictional force between the tooth surfaces, each tooth on the two half-couplings is a cantilever beam structure.

[0161] During the gear meshing process, the normal force F acting on a single tooth surface n is decomposed into a circumferential force and a radial force perpendicular to each other. Among them, the circumferential force F t is tangent to the pitch circle of the gear, and the radial force F r then points vertically downward towards the axis, and their magnitudes are respectively:

[0162]

[0163] Let T be the magnitude of the torque transmitted by the coupling in the rotor system, R be the radius of the pitch circle of the gear, and α be the magnitude of the pressure angle of the pitch circle. When a misalignment fault of the toothed coupling occurs in the rotor system, based on the misalignment offset, the inner and outer gears are most closely meshed at the upper endpoint of the coupling. Let the torque transmitted at this point be dT and the meshing angle be α 1 , and the tooth surface normal force dF at this meshing point is obtained n as:

[0164]

[0165] The magnitude of the tooth surface radial force is:

[0166]

[0167] Similarly, let the magnitude of the meshing angle at the meshing point 2 be α2 , then the magnitude of the radial force on its tooth surface is:

[0168]

[0169] dT 1 much larger than dT 2 , then dF r1 >>dF r2 , the magnitude of the radial force at other meshing points is between dF r1 and dF r2 . In the case of good alignment of the coupling, the movement centers of the left and right half couplings coincide with the movement center of the coupling sleeve at a point. Assuming that the distances between the root circles and the meshing circles of the left and right half couplings are both c at this time, the magnitudes of the radial forces received at each meshing point are equal, pointing from the center of the circle to the coupling sleeve, as Figure 2 shown; when the toothed coupling is misaligned, there is a difference in the distances between the root circles and the meshing circles of the teeth on the left and right half couplings, as Figure 3 shown;

[0170] When the misalignment offset of the coupling is △y, the minimum clearance between the coupling sleeve and the half coupling is at the upper end of the gear, and its magnitude is: C min = c - △y, the maximum clearance at the lower end is C max = c + △y, the tooth gap near the upper left side is C 2 = c - △ycosθ 1 , the tooth gap near the lower right side is C 2 ’ = c + △ycosθ 1 . By analogy, the tooth gap expressions of each tooth are obtained, and its magnitude is related to the misalignment amount △y, the tooth gap c under alignment, and the angle θ i of each tooth.

[0171] Assume that the number of teeth of the toothed coupling is z. Starting from the tooth at the uppermost end, it is sequentially numbered 1, 2, 3... z / 2 + 1, z / 2 + 2…z in the counterclockwise direction, and the angle θ i of each tooth is: θ i = 2πi / z. Each tooth is regarded as a cantilever beam. The position of the axis at rest is y, and the position of the deformed axis during the meshing process is y'. As Figure 4 shown, the deflection magnitudes at each meshing point are equal, that is, it satisfies v 1 = v 2 = … = v i = … = v z . Therefore, at the 1st tooth, we get:

[0172] M 1 (y) = F t1 [(C - Δe) - y] Equation 8

[0173] EIv″ = F t1 [(C - Δe) - y], Equation 9

[0174] Where M 1 is the bending moment of the first tooth at the meshing point, F t1 is the circumferential force of the first tooth at the meshing point, EI represents the flexural rigidity of the first tooth, v” is the second derivative of the deflection at the position of the first tooth, and integrating it gives:

[0175] According to the boundary conditions of the fixed end, we get:

[0176]

[0177] According to the boundary conditions of the fixed end, we get:

[0178]

[0179] Then we have:

[0180] The deflection of the gear coupling at the first tooth is:

[0181]

[0182] The deflection at the meshing point of the second tooth is:

[0183]

[0184] According to the deformation compatibility condition, the deflections of each tooth are the same, so:

[0185]

[0186] It is deduced that:

[0187] Without considering the change in the lever arm caused by the misalignment of the gear coupling and assuming it is always equal to d / 2, the total torque is the sum of the torques transmitted by each tooth, and its expression is:

[0188]

[0189] Let:

[0190]

[0191] Then we have: F t1 = 2T / dgE, substituting it in, the tangential force on the Nth tooth is:

[0192]

[0193] Then the radial force on the Nth tooth of the coupling is:

[0194]

[0195] Decompose F rN along the x-axis to obtain the radial force formed due to unequal torque transmission, and its magnitude is:

[0196]

[0197] Its direction is from the point where the combination is the tightest (the meshing point of tooth No. 1) to the point where the combination is the loosest (the meshing point of tooth z / 2 + 1) during the gear meshing process.

[0198] (III) Movement and force laws of the coupling

[0199] Conduct a dynamic analysis on the forms and motion states of the alignment faults of the gear coupling to obtain the motion trajectories and characteristics of the left and right half couplings and the coupling sleeve under different conditions. By establishing a radial force model generated during the coupling meshing process, obtain its distribution at each tooth and the variation laws under different misalignments and torques, including:

[0200] (1) When the misaligned gear coupling is in a normal motion state, the frequency of the coupling sleeve is equal to twice the rotational frequency of the rotor; when the gear coupling is locked due to not meeting the rotational conditions, the motion frequency of its sleeve is the same as the frequency of the rotor system;

[0201] (2) When there is a parallel misalignment fault state in the gear coupling, the phase difference of the end faces perpendicular to the axis of the left and right half couplings is 0 degrees, while in the angular misalignment fault state, the phase difference between the two is 180 degrees, and in the comprehensive misalignment fault state, the phase difference is between the two. Based on the known 2x frequency characteristics, distinguish the types of alignment faults existing in the coupling in the rotor system;

[0202] (3) When the coupling is in a parallel alignment fault state, the motion trajectory of the axis of the coupling sleeve is a cylinder, while when it is in an angular alignment fault state, the motion trajectory is a double-cone curve, and the range of motion is related to the misalignment offset. In the comprehensive alignment fault state, the motion shape of the axis of the coupling sleeve is between a cylinder and a double-cone, which is a semi-double-cone;

[0203] (4) Under the alignment fault of the gear coupling, the excitation force exerted on the rotor system during the rotation of the coupling sleeve is not only related to its mass, but also related to the misalignment offset; the magnitude of the excitation force is (2ω) 2 times the change in the frequency of the rotor system and is 4 times the excitation force generated by imbalance. For a rotor system at high speeds, the safety hazards caused by coupling misalignment are greater;

[0204] (5) The distribution laws of the radial force and circumferential force generated during the meshing process of the internal and external teeth of the gear coupling are the same at each tooth surface meshing point. They both reach the maximum value at the upper end of the coupling, that is, at the place where the gear meshing is the tightest during the meshing process, and reach the minimum value at the loosest place. At the same time, it is obtained that at the same meshing point, the value of the circumferential force is greater than that of the radial force;

[0205] (6) As the misalignment amount increases, the maximum value of the radial force increases, the minimum value decreases, and the change rate of the radial force near the upper meshing point is greater; as the torque increases, both the maximum and minimum values of the radial force increase, and the change rate of the radial force near the upper meshing point relatively increases. The change in the radial force caused by the misalignment amount is greater than that caused by the torque, which generates greater vibration and impact on the rotor system.

[0206] II. Misalignment Fault Model of the Gear Coupling in the Rotor System

[0207] The key points of rotor dynamics are the stability problem, critical speed problem, and transient response problem at different positions under the motion state of the rotor system. The traditional analysis method uses the transfer matrix method. When the operation frequency is high, the calculation accuracy will decrease, affecting the numerical stability, and the effect on ensuring the integrity of the rotor system model and the accuracy of the solution analysis is poor. Therefore, this application uses the finite element analysis method to solve the rotor dynamics problem, converting the overall analysis into segmented analysis, so that the numerical method and the analytical method are combined with each other, and the integrity of the model and the efficiency of the calculation are well balanced.

[0208] The rotor system includes discrete impellers, shaft segments, and supports. The system is discretized along the axis direction and simplified into a model composed of rigid discs, shaft segments, and bearing seats connected by nodes. The center of the disc, the center of the journal, the center of the bearing seat, and a certain point on the axis can all be selected as nodes and numbered in sequence, then the discretized rotor system model is obtained.

[0209] Establish the motion equation of the rotor system when the gear coupling has a misalignment fault state, as shown in Equation 22:

[0210]

[0211] Among them, [M] is the overall mass matrix of the system, including the rotational mass matrix and the translational inertia matrix; [C] represents the overall damping matrix of the system, including the damping matrix of the system and the gyroscopic force matrix; [K] represents the overall stiffness matrix of the rotor system; z, respectively represent the displacement value, velocity value, and acceleration value at the position of the node after the system is discretized; F r (z,t) represents the misalignment excitation force matrix when the gear coupling has a misalignment offset; is the non - linear oil film force array, Q(t) is the unbalance force array of the system; G is the gravity of the system, Q(t), F r (z, t), and G is the exciting force array received by the rotor system of the gear coupling in the misalignment fault state.

[0212] (1) Radial excitation force model of misaligned coupling meshing

[0213] As Figure 5 shown, o 1 , o 2 are respectively the rotation centers of the left and right half couplings. During the meshing process, due to the difference in the torque transmitted by each tooth between the left and right half couplings, the magnitudes of the radial forces generated are F r合 , F R合 , o 1 , o 2 are not at the same point, and F r合 points from o 1 to o', F R合 points from o 2 to o', and its motion state is as Figure 5 shown;

[0214] At a certain moment when the system is in misaligned motion, let the coordinates of the rotation center o 1 of the left half coupling be (x i , y i ), and the coordinates of the rotation center o 2 of the right half coupling be (x i+1 , y i+1 ), then there are:

[0215]

[0216] The positional relationship of each part of the gear coupling is as Figure 6 shown, where e 1 , e 2 respectively represent the distances from the rotation centers o 1 , o 2 of the left and right half couplings to the rotation center o' of the coupling sleeve during motion, then there are:

[0217] e 1 = o 1 o′ = Δy sinωt Equation 24

[0218] e 2 = o 2 o′ = Δy cosωt Equation 25

[0219] The components of the radial force F r合 of the left half coupling on the x and y axes are respectively:

[0220] F rx=-F r合 (e1)cosωt Equation 26

[0221] F ry =-F r合 (e 1 )sinωt Equation 27

[0222] For the components of the radial force F of the right half coupling on the x and y axes, there are:

[0223]

[0224] The misaligned coupling meshing radial excitation force model is obtained.

[0225] (2) Misaligned coupling sleeve centrifugal excitation force model

[0226] In the misaligned fault state, the rotation frequency of the coupling sleeve is twice the frequency of the rotor system. Then, the components f rx 、f ry of the centrifugal force exerted by the sleeve movement on the left half coupling on the x and y axes are respectively:

[0227] f rx =-m c e 1 (2ω) 2 cos(ωt) Equation 30

[0228] f ry =-m c e 1 (2ω) 2 sin(ωt) Equation 31

[0229] The components f Rx 、f Ry of the centrifugal force exerted by the right half coupling under the action of the coupling sleeve on the x and y axes are:

[0230] f Rx =-m c e 2 (2ω) 2 cos(ωt) Equation 32

[0231] f Ry =-m c e 2 (2ω) 2 sin(ωt) Equation 33

[0232] The excitation force of the misaligned toothed coupling on the rotor system is the sum of the radial force generated by torsion and the centrifugal force generated by the eccentricity of the coupling sleeve. That is, the magnitude of the misalignment force on the left and right half couplings is:

[0233] Fxi = F rx + f rx

[0234] F yi = F ry + f ry Equation 34

[0235] F x(i+1) = F Rx + f Rx

[0236] F y(i+1) = F Ry + f Ry Equation 35

[0237] where i and i + 1 represent the positions of the left and right half couplings at the nodes in the discrete system, then:

[0238]

[0239] The above expression form of the misalignment excitation force in the motion differential equation of the rotor system.

[0240] (3) Rotor unbalance excitation force model

[0241] In the rotor system, the mass eccentricity of the disk will apply an unbalance excitation force to the system during the motion. Let its initial eccentricity phase be The eccentricity of the disk is e, and the mass of the disk is m p , when the rotational speed of the rotor is ω, the component forces Q x , Q y on the x and y axes of the unbalance force are respectively:

[0242]

[0243] The expression form of the unbalance excitation force in the system vibration equation is as follows:

[0244]

[0245] where i represents the node position of the disk in the discrete system.

[0246] (4) Matrix model of the rotor system

[0247] As Figure 7 shown, the misaligned rotor system of the toothed coupling is composed of discrete elastic shaft segment units, and its generalized coordinates are the displacements at both ends of the shaft segment. Ignoring its axial deformation, its generalized coordinates are:

[0248]

[0249] where x and y are the displacements at the axis center position, and θ x , θ y is the deflection angle of the cross-section, both of which are functions of position s and time t. The displacement at any cross-section is transformed using the displacement difference function [N], where [N] = [N 1 (s), N 2 (s), N 3 (s), N 4 (s)]. Then the expression for the corresponding displacement is:

[0250] x(s, t) = [N]{u 1s} Equation 41

[0251] where [N] = [N 1 (s), N 2 (s), N 3 (s), N 4 (s)]. Then the expression for the corresponding displacement is:

[0252]

[0253] According to the boundary conditions at the endpoints, we get:

[0254]

[0255] Then the constraints of the interpolation function are:

[0256]

[0257] Assume the above interpolation function to be in the form of a cubic polynomial with respect to position s:

[0258] N 1 (s) = a 0 + a 1 s + a 2 s 2 + a 3 s 3 Equation 45

[0259] Similarly, we get:

[0260] y(s, t) = [N]{u 2s} Equation 46

[0261] The displacements at the positions of each cross-section in this element are:

[0262]

[0263] The mass per unit length of the shaft segment element is u, and its polar moment of inertia is j v , and the magnitude of the diametral moment of inertia is j d, for an infinitesimal element with an axial distance of s from node A and a thickness of ds, the expression for its bending elastic potential energy is obtained:

[0264]

[0265] Integrating it step by step along the shaft segment element with a radius of r and a length of 1, the kinetic energy and potential energy of this shaft segment can be obtained as:

[0266]

[0267] Obtained: J ps = j ps 1, where:

[0268]

[0269] [M sT is the translational inertia matrix of the element, [M sR is the rotational inertia matrix of the element, [K s is the stiffness mass matrix of the element, where the gyroscopic force matrix is [G s J = ω[J s , the mass matrix [M s of the element is a consistent mass matrix considering the rotational inertia and translational inertia of the shaft segment, that is: [M s = [M sT + [M sR ;

[0270] When the discretized rotor system is composed of N nodes and N - 1 shaft segments connected together, its overall mass matrix [M], overall stiffness matrix [K], and overall gyroscopic force matrix [G] can all be obtained by superimposing the element mass matrix, unit stiffness matrix, and unit gyroscopic matrix, and they are all matrices of order 4N×4N.

[0271] (V) Numerical solution method of the rotor system motion equation

[0272] Find the displacement values, velocities, and acceleration values generated at the node positions under the action of various excitation forces of the faulty rotor system. The solution steps are as follows:

[0273] x’ t+Δt = x’ t + (1 - γ)Δtx” t + γΔtx’ t+Δt Equation 52

[0274]

[0275] Among them, γ and β are parameters adjusted according to the requirements of integration accuracy and stability. If γ = 0.5 and β = 1 / 6 are taken, the above formula is the linear acceleration method. Assuming 0 ≤ τ ≤ △t, the integral of the acceleration expression under the linear hypothesis within the time interval △t is obtained:

[0276] (1) Calculate the initial values

[0277] ① The overall mass matrix [M], overall stiffness matrix [K], and overall damping matrix [C] of the system are obtained by superimposing the element mass matrix, element stiffness matrix, element gyroscopic force matrix, and element damping matrix of the shaft segment;

[0278] ② Given the initial displacement x, initial velocity x', and initial acceleration x'' values of the rotor system;

[0279] ③ Determine the calculation time step △t and parameters γ, β, and obtain the integration constants:

[0280] ④ Obtain the effective stiffness matrix:

[0281] ⑤ Decompose the matrix

[0282] (2) Solve for each t + △t: the equivalent excitation force, displacement, velocity, and acceleration.

[0283] (6) Establish a misalignment fault model for the rotor system of the toothed coupling

[0284] (1) Construct a mathematical model of the misalignment excitation force of the rotor system. Apply the finite element method to discretize the system, determine the vibration equation of the faulty rotor system, and based on the structural characteristics of the toothed coupling, establish the composition of the misalignment excitation force and its expression form in the vibration equation. At the same time, considering the oil film force, unbalance excitation force, and gravity, establish the corresponding mathematical model and its specific expression form in the vibration equation.

[0285] (2) By deriving the motion state of the unit shaft segment, obtain the micro-element energy equation and the magnitudes of the corresponding element mass matrix, stiffness matrix, and gyroscopic force matrix. In the discrete system composed of N nodes and N - 1 shaft segments, superimpose them in a certain form to obtain the overall mass matrix [M], stiffness matrix [K], and damping matrix [C] of the system. The number of degrees of freedom at each node position is 4, and the matrix finally presents as a 4N × 4N-order symmetric matrix.

[0286] (3) Analyze the numerical analysis method for solving the motion differential equation of the nonlinear system and solve the motion equation of the rotor system.

[0287] III. Model of the faulty rotor system of the toothed coupling

[0288] The finite element analysis method is used to discretize the actual rotor system, which is divided into 20 segments and 21 nodes. The structural parameters of the corresponding parts are set and calculated. The numerical integration is used to solve the motion differential equation of the divided rotor system, and the magnitudes of the displacement, velocity, and acceleration values at each node position are obtained. Through the vibration response analysis method of the shaft center orbit diagram, the vibration conditions of the rotor system in the healthy state and under the misalignment fault are modeled at specific positions, and the laws of their changes with system parameters such as system speed and misalignment offset are obtained.

[0289] (1) When the rotor system is well-aligned, the shaft center orbit is elliptical, the time-domain waveform is a sine wave curve, and there is only a fundamental frequency component on the frequency spectrum diagram, and the system is affected by the unbalanced force. When the toothed coupling is misaligned, the shaft center orbit is banana-shaped or in an inner 8 shape. When the misalignment amount is small (the speed is large), it is banana-shaped, and when the misalignment amount is large (the speed is small), it is in an inner 8 shape, that is, its deformation degree is affected by the misalignment amount and the speed. The time-domain waveform diagram is a double-peak waveform curve, and a second harmonic component appears on the frequency spectrum diagram in addition to the fundamental frequency. The misalignment amount and torque affect the magnitude of the misalignment excitation force of the toothed coupling, and the speed not only affects the magnitudes of the misalignment force and the unbalanced force, but also changes the distribution direction of the misalignment force in the system.

[0290] (2) The variation of the vibration response at different positions of the shafting with system parameters: The amplitudes of the fundamental frequency and second harmonic components are larger at positions closer to the coupling. When the toothed coupling is misaligned, the vibration phenomenon near it is obvious. As the misalignment offset increases, the amplitude of the fundamental frequency component remains unchanged, and the amplitude of the second harmonic component increases continuously. The misalignment amount affects the magnitude of the misalignment force and has no effect on the unbalanced force. Comparing the frequency spectrum changes on both sides of the coupling, the amplitude of the fundamental frequency component on the left side of the shafting is smaller, and the amplitude of the second harmonic component is larger. The unbalanced force on the side with a smaller shaft mass is relatively smaller, and the misalignment force is relatively larger.

[0291] (3) The stability of the rotor system is closely related to the speed. The motion state of the system is affected by the combined action of the unbalanced force, misalignment force, and oil film force, and they alternately or coupledly affect the system stability. The motion law of the system response characteristics changing with the speed is stable periodic 1 motion - periodic 2 motion - periodic 1 motion - amplitude jump - periodic 1 motion - quasi-periodic motion - periodic 3 motion - quasi-periodic motion.

Claims

1. A dynamic model of rotor misalignment fault in a rotating machinery gear coupling, characterized in that: Analyze the motion state and force conditions of the coupling under misalignment, obtain the radial force changes caused by different torque transmission during gear meshing, and consider the excitation force applied to the system by the centrifugal effect of the coupling sleeve movement process, obtain the excitation force expression of the misalignment force applied by the gear coupling to the rotor system, and establish the mathematical model of the misalignment force when the coupling has misalignment failure; Based on the effects of oil film force, unbalanced force and gravity, the dynamic vibration equation of the rotor system is established, the rotor system is discretized, the overall unit stiffness matrix, mass matrix and damping matrix of the shaft system are solved, and the motion differential equation of the rotor system is derived; the differential equation of the rotor system is solved to obtain the displacement value, velocity value and acceleration value of each node at each time, and the axis trajectory diagram, time domain waveform diagram, spectrum diagram and poincare cross-section diagram are drawn to analyze the dynamic behavior of the rotor system with coupling misalignment, as well as the effects of misalignment, speed and torque system parameters on its motion characteristics, and the stability of the system at different speeds is analyzed. The rules include: 1) At the same meshing point, the circumferential force is greater than the radial force; as the misalignment increases, the maximum value of the radial force increases, while the minimum value decreases relatively. The greater the misalignment, the greater the rate of change of the radial force at the meshing point near the upper end of the coupling, that is, the faster the meshing force of the coupling changes; when the transmitted torque value is greater, the greater the maximum value of the radial force on the coupling, and the greater the minimum value, the rate of change of the radial force near the upper end of the coupling is relatively small, that is, the impact and vibration caused by the change of the misalignment of the coupling on the system is greater than the impact of the torque change; 2) When the rotor system is well-aligned, the axis trajectory is elliptical, the time domain waveform is a sine wave curve, and there is only a 1-fold frequency component on the spectrum diagram, and the system is affected by the unbalanced force; when the gear coupling is poorly aligned, the axis trajectory is banana-shaped or inner 8-shaped, and it is banana-shaped when the misalignment is small (high speed), and it is inner 8-shaped when the misalignment is large (low speed), that is, its deformation degree is affected by the misalignment and speed. The time domain waveform is a double-peaked waveform curve, and the spectrum diagram has a 2-fold frequency component in addition to the 1-fold frequency; the misalignment and torque affect the magnitude of the misalignment excitation force of the gear coupling, and the speed not only affects the magnitude of the misalignment force and the unbalanced force, but also changes the distribution direction of the misalignment force in the system; 3) The vibration response at different positions of the shaft system changes with the system parameters: the closer to the coupling, the larger the amplitude of the 1st and 2nd frequency components. When the gear coupling is poorly aligned, the vibration nearby is more obvious; as the misalignment offset increases, the amplitude of the 1st frequency component remains unchanged, while the amplitude of the 2nd frequency component continues to increase. The misalignment affects the magnitude of the misalignment force but has no effect on the unbalanced force; the amplitude of the 1st frequency component on the left side of the shaft system is smaller, while the amplitude of the 2nd frequency component is larger. On the side with smaller shaft mass, the unbalanced force is relatively small, while the misaligned force is relatively large.

2. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1 is characterized in that: Analysis of gear coupling motion form: When the axes of the two half couplings are parallel to the design axis or at a certain angle to it, it is considered that the coupling has misalignment failure. If the coupling has misalignment failure, there are parallel misalignment, angular misalignment, and mixed misalignment failures including parallel and angular misalignment. Enlarge the trajectory diagram of the rotation center part, where o1 represents the rotation center of the left half coupling, o2 represents the rotation center of the right half coupling, o' represents the center of the coupling sleeve in motion, and o' under the centering fault is a circle with o as the center and △y as the radius. In the isosceles triangle △o1oc, the geometric relationship is obtained: ω c t=2∠o′o1o=2ωt, that is, ω c =2ω, the rotation frequency of the coupling sleeve is twice the system frequency; Assume X and Y are the motion coordinates of o' in the coordinate system, and its expression is: Similarly, the motion trajectory of its instantaneous center c' is: The motion equation of the coupling meshing circle is: (X - Δy sin 2ωt) 2 +(Y - Δy cos 2ωt) 2 =R i 2 Equation 3 When R i When R1, R2, and R are taken accordingly, they represent the trajectories formed by the inner meshing circle, the outer meshing circle, and the pitch circle during the movement of the gear coupling.

3. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1 is characterized in that: Analysis of radial force caused by misalignment: During the gear meshing process, the normal force F acting on a single tooth surface n Decomposed into perpendicular circumferential force and radial force, where the circumferential force F t Tangent to the meshing circle of the gear, radial force F r Pointing vertically downward to the axis, the sizes are: Assume that T is the torque transmitted by the coupling in the rotor system, R is the radius of the gear meshing circle, and α is the pressure angle of the meshing circle. When the gear coupling fails to align in the rotor system, based on the misalignment offset, the internal and external gears mesh most tightly at the upper end of the coupling. Assume that the torque transmitted at this point is dT, and the meshing angle is α1. The normal force dF on the tooth surface at this meshing point is obtained. n for: The radial force on the tooth surface is: Similarly, if the meshing angle at meshing point 2 is α2, the radial force on the tooth surface is: dT1 is much greater than dT2, then dF r1 >>dF r2 , the radial force at other meshing points is between dF r1 and dF r2 When the coupling is well aligned, the motion centers of the left and right half couplings coincide with the motion centers of the coupling sleeve at one point. Assuming that the spacing between the tooth root circle and the meshing circle of the left and right half couplings is c, the radial force at each meshing point is equal and points from the center of the circle to the coupling sleeve. When the coupling misalignment is Δy, the minimum clearance between the coupling sleeve and the half coupling is at the upper end of the gear, and its size is: C min =c-Δy, the maximum gap at the lower end is C max =c+Δy, the tooth gap near the upper left side is C2=c-Δycosθ1, the tooth gap near the lower right side is C2'=c+Δycosθ1, and so on to get the expression of the tooth gap between each tooth, its size is related to the misalignment Δy, the tooth gap c under the centering, and the angle θ of each tooth i association; Assume the number of teeth of the gear coupling is z, starting from the top tooth, record them in counterclockwise direction as 1, 2, 3...z / 2+1, z / 2+2...z teeth, and the angle θ of each tooth is i is: i =2πi / z, each tooth is regarded as a cantilever beam, the axis position in static state is y, the axis position in the process of meshing deformation is y', the deflection at each meshing point is equal, that is, v1=v2=…=v i =…=v z , so at tooth 1, we get: M1(y) = F t1 [(C - Δe) - y] Equation 8 EIv″ = F t1 [(C - Δe) - y] Equation 9 Where M1 is the bending moment of tooth No. 1 at the meshing point, F t1 is the circumferential force of tooth No. 1 at the meshing point, EI represents the bending stiffness of tooth No. 1, v” is the second-order derivative of the deflection at the position of tooth No. 1, and the integral is: According to the boundary conditions at the fixed end, we get: According to the boundary conditions at the fixed end, we get: Then we have: The deflection of the gear coupling at tooth No. 1 is: The deflection at the meshing point of tooth 2 is: According to the deformation coordination condition, the deflection of each tooth is the same, then: roll out: Without considering the change of the force arm caused by the misalignment of the gear coupling, it is assumed that it is always equal to d / 2. The total torque is the sum of the torques transmitted by each tooth, and its expression is: make: Then we have: F t1 =2T / dgE, substituting it into the equation, the tangential force on the Nth tooth is: Then the radial force on the Nth tooth of the coupling is: F rN Decomposed along the x-axis, the radial force generated by the unequal transmission of torque is obtained, and its magnitude is: The direction is from the tightest point to the loosest point during gear meshing.

4. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1 is characterized in that: Coupling motion force law: The form and motion state of the gear coupling misalignment failure are dynamically analyzed to obtain the motion trajectory and characteristics of the left and right half couplings and the coupling sleeve under different conditions. By establishing a radial force model generated during the coupling meshing process, its distribution at each tooth and its change law under different misalignment and torque are obtained, including: 1) When the misaligned gear coupling is in normal motion, the frequency of the coupling sleeve is equal to twice the rotor rotation frequency; when the gear coupling does not meet the rotation conditions and is locked, the movement frequency of its sleeve is the same as the frequency of the rotor system; 2) When the gear coupling has a parallel misalignment fault state, the phase difference between the left and right half couplings perpendicular to the axis is 0 degrees, while in the angular misalignment fault state, the phase difference between the two is 180 degrees. The phase difference in the comprehensive misalignment fault state is between the two. Based on the known 2-fold frequency characteristics, the type of misalignment fault in the coupling of the rotor system can be distinguished; 3) When the coupling is in a parallel misalignment fault state, the motion trajectory of the coupling sleeve axis is a cylinder, and when it is in an angular misalignment fault, the motion trajectory is a double cone curve, and the range of motion is related to the misalignment offset. In the comprehensive misalignment fault state, the motion shape of the coupling sleeve axis is between a cylinder and a double cone, which is a semi-double cone.

5. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1 is characterized in that: Rotor system gear coupling misalignment fault model: The rotor system includes discrete impellers, shaft segments and supports. The system is discretized along the axial direction and simplified into a model consisting of a rigid disk, a shaft segment and a bearing seat connected by nodes. The center of the disk, the center of the journal, the center of the bearing seat and a point on the axis line can be selected as nodes and numbered in sequence to obtain the discretized rotor system model. The motion equation of the rotor system when the gear coupling has a misalignment fault state is established, as shown in Equation 22: Where [M] is the overall mass matrix of the system, which includes the rotational mass matrix and the moving inertia matrix; [C] represents the overall damping matrix of the system, which includes the damping matrix of the system and the gyroscopic force matrix; [K] represents the overall stiffness matrix of the rotor system; z, Respectively represent the displacement value, velocity value and acceleration value at the node position after the system is discretized; F r (z, t) represents the misalignment excitation force matrix when there is misalignment offset in the gear coupling; is the nonlinear oil film force array, Q(t) is the unbalanced force array of the system; G is the gravity of the system, Q(t), F r The sum of (z, t) and G is the exciting force array that the rotor system is subjected to when the gear coupling is in the state of misalignment.

6. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1 is characterized in that: Radial excitation force model of misaligned coupling meshing: o1 and o2 are the rotation centers of the left and right half couplings respectively. Assume that the radial forces generated by the left and right half couplings during meshing are different due to the torque transmitted by each tooth. r合 、F R合 , o1 and o2 are not at the same point, and F r合 From o1 to o', F R合 From o2 to o'; At a certain moment when the system is in misalignment motion, let the coordinates of the rotation center o1 of the left half coupling be (x i ,y i ), the coordinates of the rotation center o2 of the right half coupling are (x i+1 ,y i+1 ), then: e1 and e2 represent the distances from the rotation centers o1 and o2 of the left and right half couplings to the rotation center o' of the coupling sleeve when it moves, respectively. Then: e1=o1o′=Δy sinωt Equation 24 e2=o2o′=Δy cosωt Equation 25 Radial force F of the left half coupling r合 The components on the x and y axes are: F rx =-F r合 (e1)cosωt Equation 26 F ry =-F r合 (e1)sinωt Equation 27 The components of the radial force F of the right half coupling on the x and y axes are: The radial excitation force model of misaligned coupling engagement is obtained.

7. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1 is characterized in that: Centrifugal excitation force model of misaligned coupling sleeve: In the misaligned fault state, the rotation frequency of the coupling sleeve is twice the frequency of the rotor system, then the centrifugal force applied to the left half coupling by the sleeve motion has components f on the x and y axes. rx 、f ry They are: f rx =-m c e1(2ω) 2 cos(ωt) Equation 30 f ry =-m c e1(2ω) 2 sin(ωt) Equation 31 The centrifugal force of the right half coupling on the x and y axes is f Rx 、f Ry for: f Rx =-m c e2(2ω) 2 cos(ωt) Equation 32 f Ry =-m c e2(2ω) 2 sin(ωt) Equation 33 The misalignment excitation force of the gear coupling on the rotor system is the sum of the radial force generated by the torsion and the centrifugal force generated by the eccentricity of the gear sleeve, that is, the misalignment force on the left and right half couplings is: F xi =F rx +f rx F yi =F ry +f ry Formula 34 F x(i+1) =F Rx +f Rx F y(i+1) =F Ry +f Ry Formula 35 Where i and i+1 represent the positions of the nodes of the left and right half couplings in the discrete system, then: The above expression of the misalignment excitation force in the differential equation of motion of the rotor system.

8. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1 is characterized in that: Rotor unbalanced excitation force model: The mass of the disk in the rotor system is eccentric, which will exert an unbalanced excitation force on the system during the movement. Assume that its initial eccentric phase is The eccentricity of the disk is e, and the mass of the disk is m. p When the rotor speed is ω, the unbalanced force components Q on the x and y axes are obtained. x , Q y They are: The unbalanced excitation force is expressed in the system vibration equation as follows: Where i represents the node position of the disk in the discrete system.

9. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1, characterized in that: Matrix model of rotor system: The gear coupling misaligned rotor system is composed of discrete elastic shaft segment units, and its generalized coordinates are the displacements at both ends of the shaft segment. Ignoring its axial deformation, its generalized coordinates are: Among them, x, y are the displacements at the axis position, θ x ,θ y is the deflection angle of the cross section, which is a function of position s and time t. The displacement at any cross section is converted using the displacement difference function [N], [N] = [N1(s), N2(s), N3(s), N4(s)], then the expression of the corresponding displacement is: x(s, t) = [N]{u 1s}} Equation 41 Where [N] = [N1(s), N2(s), N3(s), N4(s)], then the expression of the corresponding displacement is: According to the boundary conditions at the endpoints, we get: Then the constraints of the interpolation function are: Assume the above interpolation function to be in the form of a cubic polynomial with respect to position s: N1(s)=a0+a1s+a2s 2 +a3s 3 formula 45 Similarly, we get: y(s, t) = [N]{u 2s}} Equation 46 The displacement of each cross section in the unit is: The mass per unit length of the shaft segment is u, and its polar moment of inertia is j v , the diameter moment of inertia is j d , for a microelement with an axial distance s from node A and a thickness ds, the expression for its bending elastic potential energy is obtained: By integrating it step by step along the axis segment unit with radius r and length 1, the kinetic energy and potential energy of the axis segment can be obtained as follows: Get: J ps =j ps 1, where: [M sT ] is the unit's moving inertia matrix, [M sR ] is the unit's rotational inertia matrix, [K s ] is the stiffness mass matrix of the unit, where the gyro force matrix is ​​[G s ]J=ω[J s ], the mass matrix of the unit [M s ] is the consistent mass matrix taking into account the rotational inertia and translational inertia of the shaft segment, that is: [M s ]=[M sT ]+[M sR ]; When the discretized rotor system is composed of N nodes and N-1 shaft segments, its overall mass matrix [M], overall stiffness matrix [K], and overall gyroscopic force matrix [G] can all be formed by superimposing the unit mass matrix, unit stiffness matrix, and unit gyroscopic matrix, and they are all matrices of the order 4N×4N.

10. The dynamic model of rotor centering fault of rotating machinery gear coupling according to claim 1, characterized in that: Numerical solution of the rotor system motion equation: To obtain the displacement, velocity and acceleration values ​​generated at the node position of the faulty rotor system under the action of various excitation forces, the solution steps are as follows: x' t+Δt =X' i +(1-γ)Δtx” t +γΔtx' t+Δt formula 52 Among them, γ and β are parameters adjusted according to the accuracy and stability requirements of the integral. If γ = 0.5 and β = 1 / 6, the above formula is the linear acceleration method. Assuming 0 ≤ τ ≤ Δt, the integral of the acceleration expression with linear assumption within the interval time Δt is obtained: (1) Calculate the initial value ① The overall mass matrix [M], overall stiffness matrix [K], and overall damping matrix [C] of the system are obtained by superimposing the unit mass matrix, unit stiffness matrix, unit gyroscopic force matrix, and unit damping matrix of the shaft segment; ② Given the initial displacement x, initial velocity x' and initial acceleration x" of the rotor system; ③ Determine the calculation time step Δt and parameters γ and β, and obtain the integral constant: ④Get the effective stiffness matrix: ⑤ Decomposition matrix (2) For each t+At, solve: equivalent excitation force, displacement, velocity and acceleration.

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