An analysis method for torsional vibration exciting bending self-excited vibration of a rotor

Abstract

The application relates to the technical field of aero-engine dynamics, in particular to a method for analyzing torsional vibration-induced bending self-excited vibration of a rotor, which comprises the following steps: establishing a periodic time-varying vibration equation of a rotor system, and constructing the periodic time-varying vibration equation containing a static displacement term; based on the perturbation method and the harmonic balance method, the periodic time-varying vibration equation containing the static displacement term is solved to obtain multi-order approximate responses of the rotor system under torsional vibration excitation; based on whether there is a time-growing term in each order approximate response, an instability criterion for the bending self-excited vibration of the rotor system is determined; based on the instability criterion, the stability of the rotor system is evaluated, dangerous working conditions in which the bending self-excited vibration may occur are predicted, and corresponding avoidance measures are determined based on the evaluation results. The method can reveal the instability mechanism and give an accurate instability condition criterion, thereby providing a reliable theoretical basis for the dynamic design and fault prevention of rotating machinery.

Description

Technical Field

[0001] The embodiments of this application relate to the field of aero-engine dynamics technology, and in particular to an analysis method for torsional vibration-induced rotor bending self-excited vibration. Background Technology

[0002] High-speed rotating machinery, such as aircraft engines, experiences both bending and torsional vibrations in their rotors during operation. Traditional vibration analysis methods typically decouple these two vibration modes: assuming a constant rotational speed in bending vibration analysis and ignoring the influence of bending vibration in torsional vibration analysis. This linear decoupling method is applicable to lower speeds or smaller torsional amplitudes. However, under high-speed, high-excitation conditions, the coupling effect between bending and torsional vibrations becomes significant. Studies show that torsional vibration of the rotor, through the gyroscopic effect, causes the rotor system's stiffness to become time-varying, transforming the entire system into a periodic time-varying system. The direct consequence is that, under certain conditions, torsional vibration can continuously input energy into bending vibration, triggering self-excited vibration in the bending direction, ultimately leading to system instability. This instability manifests as asynchronous vibration, generating alternating bending stress on the shaft, which can, in severe cases, lead to fatigue fracture of the shaft, posing a significant threat to equipment safety.

[0003] Several analytical methods considering bending-torsional coupling already exist in the prior art. For example, patent document CN115876450A discloses a method for calculating shaft system response considering gyro-torsional coupling, which constructs a nonlinear mathematical model incorporating gyroscopic effects and hydraulic excitation, thus considering more vibration coupling factors. Patent document CN113029322B proposes a method for calculating longitudinal-torsional coupling vibration of marine diesel engine shaft systems, based on the deformation energy method to calculate coupling stiffness and vibration response.

[0004] However, existing technologies still have the following shortcomings: First, most analysis methods over-linearize time-varying systems, failing to fully consider the influence of gyro torque changes over time, leading to significant errors between the calculated results and actual conditions. Second, while existing stability analysis methods (such as the Flokai method and numerical methods) can provide numerical conclusions, their physical meaning is unclear, failing to intuitively reveal the intrinsic mechanism of instability and the influence of various parameters. Third, existing analyses fail to fully consider the significant impact of static displacement of the rotor due to factors such as gravity on bending-torsional coupled self-excited vibration. Finally, there is a lack of a systematic set of instability criterion criteria for multi-harmonic torsional vibration excitation. Summary of the Invention

[0005] In view of this, embodiments of this application propose an analytical method for torsional vibration-induced rotor bending self-excited vibration. This method aims to solve the problem of inaccurate predictions caused by neglecting the time-varying nature of gyroscopic torque, unclear physical mechanisms, and lack of systematic criteria when analyzing torsional vibration-induced rotor bending self-excited vibration. By establishing a time-varying model containing static displacement and solving it using an improved perturbation method, the instability mechanism is revealed, and accurate instability condition criteria are given, providing a reliable theoretical basis for the dynamic design and fault prevention of rotating machinery.

[0006] To achieve the above objectives, embodiments of this application propose an analysis method for torsional vibration-induced rotor bending self-excited vibration, the method comprising the following steps: Establish the periodic time-varying vibration equation of the rotor system; A static displacement term generated by gravity, inertial force, and inertial torque is introduced into the periodic time-varying vibration equation to construct a periodic time-varying vibration equation that includes the static displacement term. Based on the perturbation method and harmonic balance method, the periodic time-varying vibration equation containing the static displacement term is solved to obtain the multi-order approximate response of the rotor system under torsional vibration excitation. Based on whether there are terms that increase with time in each order of approximate response, the instability criterion for the rotor system to undergo bending self-excited vibration is determined; Based on the instability criterion, the stability of the rotor system is assessed to predict the dangerous operating conditions that may cause bending self-excited vibration, and based on the assessment results, corresponding avoidance measures are determined.

[0007] To achieve the above objectives, embodiments of this application also propose an analysis system for torsional vibration-induced rotor bending self-excited vibration, the system comprising: The equation building module is used to establish the periodic time-varying vibration equation of the rotor system. The static displacement processing module is used to introduce static displacement terms generated by gravity, inertial force and inertial torque into the periodic time-varying vibration equation to construct a periodic time-varying vibration equation containing static displacement terms. The vibration response solution module is used to solve the periodic time-varying vibration equation containing static displacement terms based on the perturbation method and the harmonic balance method, so as to obtain the multi-order approximate response of the rotor system under torsional vibration excitation. The instability criterion determination module is used to determine the instability criterion for the rotor system to undergo bending self-excited vibration based on whether there is a term that increases with time in each order of approximate response. The stability assessment and decision-making module is used to assess the stability of the rotor system based on the instability criterion, predict dangerous operating conditions that may lead to bending self-excited vibration, and determine corresponding avoidance measures based on the assessment results.

[0008] To achieve the above objectives, embodiments of this application also propose an electronic device, including a processor and a memory, wherein the memory stores instructions executable by the processor, and the processor is configured to execute the instructions such that the electronic device can implement the analysis method for torsional vibration-induced rotor bending self-excited vibration as described above.

[0009] To achieve the above objectives, embodiments of this application also propose a computer-readable storage medium storing a computer program that, when executed by a processor, enables the implementation of an analysis method for torsional vibration-induced rotor bending self-excited vibration as described above.

[0010] This application proposes an analysis method for torsional vibration-induced bending self-excited vibration of a rotor. The method establishes a periodic time-varying vibration equation for the rotor system. Static displacement terms generated by gravity, inertial force, and inertial torque are introduced into the periodic time-varying vibration equation to construct a periodic time-varying vibration equation containing static displacement terms. Based on the perturbation method and harmonic balance method, the periodic time-varying vibration equation containing static displacement terms is solved to obtain the multi-order approximate response of the rotor system under torsional vibration excitation. Based on whether there are time-increasing terms in each order of approximate response, an instability criterion for bending self-excited vibration of the rotor system is determined. Based on the instability criterion, the rotor system is subjected to bending self-excited vibration. The subsystem undergoes stability assessment to predict dangerous conditions that may lead to bending self-excited vibration, and based on the assessment results, corresponding avoidance measures are determined. Since traditional vibration analysis linearly decouples bending and torsional vibrations, resulting in large errors under high speeds and strong excitation, this scheme establishes a periodic time-varying vibration equation (Hill equation) that incorporates gyroscopic effects and bending-torsional coupling. This allows for a complete mathematical modeling of the physical quantities by which torsional vibration causes the system stiffness to vary over time due to gyroscopic effects, overcoming the shortcomings of traditional linearization methods under strong coupling conditions. Subsequently, static displacements generated by gravity, inertial forces, and inertial moments are introduced into the periodic time-varying vibration equation. This method explicitly considers the contribution of static displacement caused by gravity and inertial forces during maneuvering to the bending-torsional coupling effect, thus significantly improving stability and computational accuracy. Then, it employs a perturbation method with torsional amplitude as a small parameter, combined with a harmonic balance method for step-by-step analytical solutions. Compared to purely numerical methods (such as the Flokai method), this approach gradually reveals how torsional vibration energy couples with static displacement and natural frequency through various harmonics and is transferred to bending vibration. It fundamentally elucidates the intrinsic physical meaning of torsional vibration exciting self-excited bending vibration, rather than merely providing numerical conclusions. Furthermore, based on the analytical solution process, this method can systematically... This method derives complete and clear instability criteria and provides targeted preventive measures. Based on this, it establishes an accurate time-varying model, considers key static displacement effects, adopts a mechanism-transparent analytical solution, derives systematic instability criteria, and forms a complete evaluation process. This solves the problem of inaccurate predictions caused by neglecting the time-varying nature of gyroscopic torque, unclear physical mechanisms, and lack of systematic criteria when analyzing rotor bending self-excited vibration induced by torsional vibration. This method combines theoretical rigor with engineering practicality and is particularly suitable for the bending-torsional coupled dynamic design and stability analysis of high-end rotating machinery such as aero-engines.

[0011] Optionally, the periodic time-varying vibration equation of the rotor system is established, including: establishing the rotor system vibration equation that includes gyroscopic effects and bending-torsional coupling, and defining the total transient rotation angle of the rotor as the sum of the average rotational speed and the torsional vibration component; expanding the torsional vibration component into a complex Fourier series containing Nth-order harmonics; substituting the torsional vibration in the form of the complex Fourier series into the rotor system vibration equation to obtain the periodic time-varying vibration equation.

[0012] Optionally, a static displacement term generated by gravity, inertial force, and inertial torque is introduced into the periodic time-varying vibration equation to construct a periodic time-varying vibration equation containing a static displacement term. This includes setting the origin of the coordinate system of the periodic time-varying vibration equation from the geometric axis to the static equilibrium position of the rotor when it does not undergo torsional vibration under the combined action of gravity, inertial force, and inertial torque; and introducing a gravity term on the right-hand side of the periodic time-varying vibration equation to consider the static displacement caused by gravity.

[0013] Optionally, based on the perturbation method and harmonic balance method, the periodic time-varying vibration equation containing the static displacement term is solved to obtain the approximate responses of the rotor system under torsional vibration excitation. This includes: expanding the solution of the periodic time-varying vibration equation containing the static displacement term into a perturbation series with the amplitude of the rotor torsional vibration as a small parameter; solving the zero-order approximate response of the perturbation series to obtain the asynchronous bending vibration response containing the static displacement caused by gravity and inertial force, as well as the vibration at the forward and reverse precession natural frequencies of the rotor; substituting the zero-order approximate response into the periodic time-varying vibration equation to solve the first-order approximate equation to obtain the response generated by the combined excitation of torsional vibration harmonics and natural frequencies; substituting the zero-order and first-order approximate responses into the periodic time-varying vibration equation to construct and solve the second-order approximate equation, and applying the harmonic balance method to solve it to obtain the second-order approximate response and further verify the instability condition.

[0014] Optionally, the criteria for rotor bending vibration instability include: a forward precession instability criterion and a reverse precession instability criterion; the criterion for determining the instability of the rotor system due to bending self-excited vibration based on whether there is a term that increases with time in each order of approximate response includes: when the rotor torsional vibration... First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experiences bending precession instability; when the rotor torsional vibration... First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experiences bending reverse precession self-excited vibration instability; when the rotor torsional vibration... First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experiences bending precession instability; when the rotor torsional vibration... First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experienced bending reverse precession self-excited vibration instability; among which, For harmonic orders, ; It is the positive precession natural frequency. It is the natural frequency of the reverse precession.

[0015] Optionally, a stability assessment of the rotor system is performed, including: calculating the forward and reverse precession natural frequencies of the rotor system at the operating speed based on the instability criterion; comparing the forward and reverse precession natural frequencies with the instability criterion to determine the torsional vibration harmonic order that causes instability and its corresponding critical speed; and determining whether instability has occurred based on the torsional vibration harmonic order that causes instability and its corresponding critical speed, combined with the actual torsional vibration amplitude and damping parameters of the rotor system.

[0016] Optionally, based on the instability criterion, the forward precession natural frequency and the reverse precession natural frequency of the rotor system at the operating speed are calculated, including: forward precession natural frequency. The characteristic equation is: ; Reverse precession natural frequency The characteristic equation is: ; in, For quality, The moment of inertia of the diameter. The moment of inertia is the polar rotation. The average rotational speed, , , , This is the stiffness coefficient; This represents the normal precession frequency. It represents the reverse precession natural frequency.

[0017] Optionally, preventive measures may include at least one of the following: adjusting the structural parameters of the rotor system so that its forward and reverse precession natural frequencies avoid the dangerous frequency conditions determined by the instability criterion; avoiding torsional resonance of the rotor through design; adding a damper to the rotor system; and for rotors with toothed connections, ensuring that the cylindrical centering surfaces at both ends of the connection are tightly fitted to reduce static displacement. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments or related technologies of this application, the accompanying drawings used in the description of the embodiments or related technologies of this application will be briefly introduced below. Obviously, the following drawings are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. The drawings described herein are only used to explain this application and are not intended to limit this application.

[0019] Figure 1 This is a flowchart of an analysis method for torsional vibration-induced rotor bending self-excited vibration provided in one embodiment of this application; Figure 2 This is a schematic diagram of the combined trajectory of forward and reverse precession trajectories and rotor shaft center motion provided in one embodiment of this application; Figure 3 This is a schematic diagram of the instability criterion provided in one embodiment of this application; Figure 4 This is a flowchart of a calculation method for torsional vibration-excited rotor bending self-excited vibration provided in one embodiment of this application; Figure 5 This is a schematic diagram of the structure of an analysis system for torsional vibration-excited rotor bending self-excited vibration provided in another embodiment of this application; Figure 6 This is a schematic diagram of the structure of an electronic device provided in another embodiment of this application. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the various embodiments of this application will be described in detail below with reference to the accompanying drawings. Those skilled in the art will understand that many technical details have been presented in the embodiments of this application to facilitate better understanding. However, the technical solutions claimed in this application can be implemented even without these technical details and various variations and modifications based on the following embodiments. The division of the following embodiments is for ease of description and should not constitute any limitation on the specific implementation of this application. The following embodiments can be combined with and referenced by each other without contradiction.

[0021] High-speed rotating machinery, such as aircraft engines, experiences both bending and torsional vibrations in their rotors during operation. Traditional vibration analysis methods typically decouple these two vibration modes: assuming a constant rotational speed in bending vibration analysis and ignoring the influence of bending vibration in torsional vibration analysis. This linear decoupling method is applicable to lower speeds or smaller torsional amplitudes. However, under high-speed, high-excitation conditions, the coupling effect between bending and torsional vibrations becomes significant. Studies show that torsional vibration of the rotor, through the gyroscopic effect, causes the rotor system's stiffness to become time-varying, transforming the entire system into a periodic time-varying system. The direct consequence is that, under certain conditions, torsional vibration can continuously input energy into bending vibration, triggering self-excited vibration in the bending direction, ultimately leading to system instability. This instability manifests as asynchronous vibration, generating alternating bending stress on the shaft, which can, in severe cases, lead to fatigue fracture of the shaft, posing a significant threat to equipment safety.

[0022] Patent document CN115876450A discloses a method for calculating shaft system response considering gyro-torsional coupling. It constructs a nonlinear mathematical model of bending-torsional coupled vibration incorporating gyroscopic effects and hydraulic excitation, considering the influence of gyroscopic effects on shaft system vibration characteristics and hydraulic excitation and its resulting coupling torque, thus solving the problem of vibration coupling factors that were not fully considered in existing technologies. Patent document CN113029322B proposes a method for calculating longitudinal-torsional coupled vibration of a marine diesel engine shaft system. Based on the deformation energy method of materials mechanics, it calculates the longitudinal-torsional coupled stiffness of the crankshaft, ultimately calculating the inherent characteristics of longitudinal-torsional coupled vibration of the ship shaft system and the forced vibration response of the ship shaft system.

[0023] However, existing technologies still have the following shortcomings: First, most analysis methods over-linearize time-varying systems, failing to fully consider the influence of gyro torque changes over time, leading to significant errors between the calculated results and actual conditions. Second, while existing stability analysis methods (such as the Flokai method and numerical methods) can provide numerical conclusions, their physical meaning is unclear, failing to intuitively reveal the intrinsic mechanism of instability and the influence of various parameters. Third, existing analyses fail to fully consider the significant impact of static displacement of the rotor due to factors such as gravity on bending-torsional coupled self-excited vibration. Finally, there is a lack of a systematic set of instability criterion criteria for multi-harmonic torsional vibration excitation.

[0024] Therefore, there is an urgent need for an analytical and computational method that can reveal the intrinsic mechanism of torsional vibration-induced rotor bending self-excited vibration and provide clear instability criteria.

[0025] In view of this, embodiments of this application propose an analytical method for torsional vibration-induced rotor bending self-excited vibration. This method aims to solve the problem of inaccurate predictions caused by neglecting the time-varying nature of gyroscopic torque, unclear physical mechanisms, and lack of systematic criteria when analyzing rotor bending self-excited vibration induced by torsional vibration in existing methods. By establishing a complete time-varying differential equation for rotor bending-torsional coupled vibration and solving it using an improved perturbation method, the instability mechanism is revealed. Accurate instability condition criteria are also given, providing a reliable theoretical basis for the dynamic design and fault prevention of rotating machinery.

[0026] One embodiment of this application proposes an analysis method for torsional vibration-induced rotor bending self-excited vibration, applied to an electronic device, wherein the electronic device can be a terminal or a server. This embodiment and the following embodiments will use a server as an example for description. The implementation details of the analysis method for torsional vibration-induced rotor bending self-excited vibration proposed in this embodiment are described in detail below. The following implementation details are provided for ease of understanding and are not necessary for implementing this solution.

[0027] The specific process of the analysis method for torsional vibration-induced rotor bending self-excited vibration proposed in this embodiment can be described as follows: Figure 1 As shown, it includes: Step 101: Establish the periodic time-varying vibration equation of the rotor system.

[0028] This step aims to construct a mathematical model that accurately reflects the bending-torsional coupled dynamic characteristics of the rotor. Specifically, it is based on the rotor bending vibration equation considering gyroscopic effects. For example... Figure 2 As shown, for a typical Jeffcott rotor model, its vibration equation can be expressed as a system of differential equations containing the mass matrix, gyroscope matrix, and stiffness matrix.

[0029] In one possible embodiment, step 101 includes: establishing a rotor system vibration equation that includes gyroscopic effects and bending-torsional coupling, and defining the total transient rotation angle of the rotor as the sum of the average rotational speed and the torsional vibration component; expanding the torsional vibration component into a complex Fourier series containing Nth-order harmonics; and substituting the torsional vibration in the form of the complex Fourier series into the rotor system vibration equation to obtain a periodic time-varying vibration equation.

[0030] For example, the vibration equation of a rotor system can be expressed as follows: ; Let the transient rotation angle of the rotor be... It is the sum of its average rotational speed and torsional vibration component: ; in, The average speed refers to the average speed of the engine while it is running. The torsional vibration of the rotor can be represented as follows: ; in, For torsional vibration waveform function, This represents the torsional amplitude.

[0031] The key improvement lies in expanding the torsional vibration component into a complex Fourier series containing the Nth harmonic: ; Substituting the time-varying terms of the gyro torque matrix and stiffness matrix, which vary with torsional vibration, into the differential equation of rotor bending vibration, we obtain the following periodic time-varying homogeneous equation, which is also the periodic time-varying vibration equation (Hill equation): ; This equation is a homogeneous periodic time-varying differential equation, and the period of its coefficient matrix is... It accurately describes how torsional vibrations, through the gyroscopic effect, transform a rotor system into a parametrically excited system.

[0032] Step 102: Introduce a static displacement term generated by gravity, inertial force and inertial torque into the periodic time-varying vibration equation to construct a periodic time-varying vibration equation containing the static displacement term.

[0033] Understandably, one of the key differences between this method and traditional analysis lies in its explicit consideration of the influence of static displacement (gravitational displacement) on bending-torsional coupled self-excited vibration. Since torsional vibration causes the system stiffness to vary over time, the static displacement caused by gravity couples with this time-varying stiffness, becoming a significant excitation source for self-excited vibration. Furthermore, during maneuvering flight, the additional inertial forces and moments change significantly, coupling with the rotational effect generated by torsional vibration, potentially exacerbating the rotor's self-excited vibration. Therefore, in the solution process, the origin of the coordinate system must be taken at the centerline position of the rotor when it is not subjected to gravity (and inertial forces).

[0034] In one possible embodiment, step 102 includes: setting the origin of the coordinate system of the periodic time-varying vibration equation from the geometric axis to the static equilibrium position of the rotor when it does not undergo torsional vibration under the combined action of gravity, inertial force and inertial torque; and introducing a gravity term on the right-hand side of the periodic time-varying vibration equation to consider the static displacement caused by gravity.

[0035] Introduce a static force vector on the right-hand side of the equation, consisting of gravity, inertial force, and inertial torque: ; in, For gravity, This represents the static moment caused by gravity. In maneuvering flight analysis, this can be expanded to an equivalent static load vector that includes additional inertial forces and gyroscopic moments. .

[0036] Therefore, the periodic time-varying vibration equation including the static displacement term is: ; This treatment ensures the static displacement caused by gravity. It can be correctly separated and considered as part of the 0th-order approximate solution in the subsequent perturbation solution.

[0037] Step 103: Based on the perturbation method and harmonic balance method, solve the periodic time-varying vibration equation containing the static displacement term to obtain the multi-order approximate response of the rotor system under torsional vibration excitation.

[0038] Understandably, there are currently several methods for solving the equations of periodic time-varying systems, including perturbation methods, harmonic balance methods, the Floquet method, and numerical methods. Among these, the perturbation method can directly solve second-order differential equations, is relatively transparent and easy to understand, and has a clear physical meaning.

[0039] For example, a multi-order approximation response may include a zero-order approximation response (zero-order approximation solution), a first-order approximation response (first-order approximation solution), and a second-order approximation response (second-order approximation solution).

[0040] It should be noted that the perturbation method provided in the embodiments of this application differs from the traditional perturbation method in that: it does not avoid the extreme conditions and the response that grows with time in the equation, but uses these as the basis to judge the instability conditions of the self-excited vibration of the system; the 0th order approximate solution must include the gravity displacement term.

[0041] In one possible embodiment, step 103 includes: expanding the solution of the periodic time-varying vibration equation containing the static displacement term into a perturbation series, using the amplitude of the rotor torsional vibration as a small parameter; solving the zero-order approximate response of the perturbation series to obtain the asynchronous bending vibration response containing the static displacement caused by gravity and inertial force, and the vibration at the forward and reverse precession natural frequencies of the rotor; substituting the zero-order approximate response into the periodic time-varying vibration equation to solve the first-order approximate equation to obtain the response generated by the combined excitation of each order of torsional vibration harmonics and natural frequencies; substituting the zero-order approximate response and the first-order approximate response into the periodic time-varying vibration equation to construct and solve the second-order approximate equation, and applying the harmonic balance method to solve it to obtain the second-order approximate response and further verify the instability condition.

[0042] Due to torsional amplitude To address the issue of small parameters, this method employs an improved perturbation method. The improvement lies in not avoiding the extreme conditions (resonance conditions) appearing in the equations, but rather using them as a basis for judging system instability. Let the solution to the equations be a small parameter... The power series form: ; In the formula, The exponent is the approximate order of the solution.

[0043] Substituting the above expansion into the periodic time-varying vibration equation obtained in step 102, let... If the coefficients of the same power are equal, then approximate equations of various orders can be obtained.

[0044] (1) 0th order approximate solution Substituting the values ​​into the equation and solving the zeroth-order homogeneous equation containing the gravity term yields the rotor's static displacement and its natural precession frequency ω. + and the natural frequency of the reverse precession ω - Solution of asynchronous bending vibration.

[0045] The complete solution of the 0th order response equation is ; This indicates that the rotor vibrates at its natural frequency, i.e., asynchronous vibration. The vibration amplitude is related to higher-order response terms.

[0046] in, It is the positive precession natural frequency. The reverse precession natural frequency is obtained from the characteristic equation. This solution shows that even without external excitation, the rotor exhibits asynchronous bending vibration at its natural frequency, determined by the initial conditions.

[0047] (2) First-order approximate solution Substituting the 0th-order approximate solution and rearranging, we obtain the excitation terms on the right-hand side of the 1st-order equation. We then solve for the responses to each excitation term. By analyzing the characteristic determinant of the response denominator, we identify the frequency conditions that may cause forward and reverse precession instability.

[0048] Substituting the 0th-order solution into the 1st-order equation, the right-hand side of the equation will show excitation terms with multiple frequency combinations. The forced response is then solved for each of these excitation terms. By analyzing the denominator of the response solution (i.e., the determinant of the dynamic stiffness matrix of the system at that frequency combination), the frequency conditions that cause the response to tend towards infinity (i.e., poles) can be identified, that is, the frequency conditions that may cause positive precession instability and negative precession instability can be identified.

[0049] The equation for the first-order approximate response is obtained by rearranging:

[0050]

[0051]

[0052] ; ① First, solve for the gravitational response, that is, the response of the first term on the right-hand side of the equation. At this point, the equation is:

[0053]

[0054] Therefore, we can conclude that: ; In the formula: ; The subscript "1" indicates a first-order approximate solution; "Indicates gravitational response;" " indicates the k-th positive precession; The subscript "1" indicates a first-order approximate solution; "Indicates gravitational response;" " indicates the k-th order anti-precession.

[0055] ② Solve for the following two terms on the right-hand side of the equation: ; ; The result is: ; ; ; ; In the formula: ; The conditions for rotor instability are: ,or ;

[0056] The above formula represents the criterion for the forward precession stability of rotor bending self-excited vibration. For example... , The first harmonic component of torsional vibration will cause the rotor to bend self-excited forward precession instability; , The second harmonic component of torsional vibration will excite the rotor to bend and self-excited forward precession instability.

[0057] ③ Incentive items Solution: ; In the formula: ; For any and The condition under which torsional vibration will cause rotor bending and reverse precession instability is: ,or (k = 1, 2, 3,…); The above equation represents the criterion for the forward precession stability of rotor bending self-excited vibration. For example, ,when At that time, the first harmonic component of rotor torsional vibration will cause rotor bending reverse precession instability; ,when At that time, the second harmonic component of the rotor's torsional vibration will cause the rotor to bend and reverse precession instability.

[0058] (3) Second-order approximate solution Substituting the 0th and 1st order solutions, the harmonic balance method is used to write out the balance equations for each order of harmonics (0th, ±1st, and ±2nd harmonics) and solve them separately. The time-growth response that occurs when the characteristic determinant of each harmonic is zero is analyzed to obtain a more complete set of instability conditions.

[0059] Understandably, to further confirm and supplement the instability criterion and analyze higher-order coupling effects, the second-order equations are solved. Substituting the 0th and 1st-order solutions, the solutions to the second-order equations exhibit more complex harmonic combinations. Using the harmonic balance method, the harmonic balance equations for each order (0th, ±1st, and ±2nd harmonics) are written out and solved separately.

[0060] The second-order approximation equation is: ; As the analysis above shows, the second-order approximate solution also includes... , , and The series form of the components can be expressed as: ; In the formula, The superscript "+" indicates The "+" symbol; the subscript "2" represents a second-order approximate solution, the subscript "+" represents positive precession, and the subscript "h" represents the "th" digit of the series. "item. Superscript “-” indicates The "-" symbol indicates a second-order approximate solution; the subscript "2" indicates a positive precession; and the subscript "h" indicates the "th"th order of the series. "item.

[0061] Substituting the second-order solution, the equation can be expressed as:

[0062] In the formula: for The coefficient vector, with the superscript "+" indicating The "+" sign; for The coefficient vector, with the superscript "-" indicating of"-".

[0063] ; ; For the equation to hold, the coefficient vector... and All must be 0. According to the harmonic balance method, the coefficients of harmonics of the same order must be 0. Therefore, the second-order approximate solution expressed by the first-order and 0th-order approximate solutions can be obtained. For clarity, let's assume... and Solve for the coefficient vectors separately, that is, solve for them separately. and Two equations.

[0064] ① ; 1) Solution to the equilibrium equation of the 0th harmonic: ; In the formula: ; 2) The equilibrium equations and solutions of the +1st harmonic, i.e., the solutions obtained with coefficients of 0. : ; In the formula: ; because Therefore, the solution to the equation is a bounded value, meaning that the +1 harmonic will not become unstable.

[0065] 3) The equilibrium equation for the -1st harmonic, i.e. The solution obtained by setting the coefficient to 0: ; In the formula: ; Therefore, when ,or hour, , As the frequency approaches infinity, torsional vibration causes the rotor to become unstable due to reverse precession, with an instability frequency of [frequency value missing]. .

[0066] Comparing the above formulas, when the torsional vibration is symmetrical, the amplitudes of the forward and reverse precession of the rotor caused by the torsional vibration are close, and the trajectory is a very flat ellipse.

[0067] 4) The equilibrium equation for the +2nd harmonic, i.e. The solution obtained by setting the coefficient to 0: ; In the formula: ; because The solution to the equation is bounded, so the rotor will not experience +2nd harmonic instability.

[0068] 5) The equilibrium equation for the second harmonic, i.e. The solution obtained by setting the coefficient to 0: ; In the formula: ; Using the same method as before, we determine the stability of the -2nd harmonic based on the above formula. When ,or hour, , As the frequency approaches infinity, the -2nd harmonic component of the torsional vibration will cause the rotor to become unstable due to reverse precession, with an instability frequency of... .

[0069] Similarly, when the torsional vibration is symmetrical, the trajectory is a very flat ellipse.

[0070] ② Solution: (1) Equilibrium equation for the 0th harmonic: ; As can be seen from the above equation, torsional vibration will cause the rotor bending reverse precession self-excited vibration to increase continuously over time, resulting in reverse precession instability.

[0071] (2) The equilibrium equation for the +1st harmonic, i.e. The solution obtained by setting the coefficient to 0: ; In the formula: ; The characteristic equation is still used to determine the stability of the +1st harmonic. When ,or hour, , As the frequency approaches infinity, the +1st harmonic component of the torsional vibration will cause positive precession instability of the rotor, with an instability frequency of... .

[0072] (3) The equilibrium equation for the -1st harmonic, i.e. The solution obtained by setting the coefficient to 0: ; In the formula: ; because Therefore, the -1st harmonic component of torsional vibration will not cause rotor precession instability.

[0073] (4) The equilibrium equation for the +2nd harmonic, i.e. The solution obtained by setting the coefficient to 0: ; In the formula: ; This allows us to determine the stability of the +2nd harmonic. ,or hour, , As the frequency approaches infinity, the +2nd harmonic component of the torsional vibration will cause positive precession instability of the rotor, with an instability frequency of... .

[0074] (5) The equilibrium equation for the -2nd harmonic, i.e. The solution obtained by setting the coefficient to 0: ; In the formula: ; because The solution to the equation is bounded, so the rotor will not experience -2nd harmonic instability.

[0075] Step 104: Based on whether there are terms that increase with time in each order of approximate response, determine the instability criterion for the rotor system to undergo bending self-excited vibration.

[0076] For example, the criteria for rotor bending vibration instability include: the forward precession instability criterion and the reverse precession instability criterion.

[0077] Based on the analysis of the 0th, 1st, and 2nd order approximate solutions in the above embodiments, particular attention is paid to the occurrence of poles (denominator is zero) or the inclusion of linearly increasing factors with time (such as...). Given the conditions, a complete set of instability criteria can be systematically derived.

[0078] Based on the results of perturbation solutions of various orders, the condition that the response contains a term that continuously increases with time (i.e., the characteristic determinant is zero) is a necessary condition for system instability. The complete instability criterion is as follows: In one possible embodiment, step 104 includes: When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At this time, the rotor system experiences bending forward precession instability; When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experienced bending reverse precession self-excited vibration instability; When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At this time, the rotor system experiences bending forward precession instability; When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experienced bending reverse precession self-excited vibration instability; in, For harmonic orders, ; It is the positive precession natural frequency. It is the natural frequency of the reverse precession.

[0079] The greater the rotor flexibility, the greater the impact of torsional vibration. That is, rotors with slender shafts are more susceptible to torsional vibration excitation and instability. For example, in a dual-rotor system, the torsional vibration excitation stability of the low-pressure rotor should be the focus.

[0080] It is understandable that, such as Figure 2 As shown, Figure 2This demonstrates the decomposition and synthesis of the rotor's axial motion under torsional vibration excitation. The forward precession trajectory is an elliptical trajectory, rotating in the same direction as the rotor's rotation. It represents the positive whirl component generated by the rotor under torsional vibration excitation, with a vibration frequency typically a combination of the rotor's natural frequency and torsional frequency. Similarly, the reverse precession trajectory is an elliptical trajectory, but rotating in the opposite direction to the rotor's rotation. It represents the reverse whirl component generated by the rotor under torsional vibration excitation. The composite trajectory (the actual trajectory of the shaft center) is formed by the vector superposition of the forward and reverse precession elliptical trajectories. When the torsional vibration is symmetrical about the equilibrium position, the amplitudes of the forward and reverse precession are similar, and the composite trajectory appears as a very flattened ellipse, even approaching a straight line. The major axis of the flattened ellipse reflects the main vibration direction of the rotor's bending vibration, while the extremely small minor axis indicates that vibrations perpendicular to this direction are significantly canceled out.

[0081] Torsional vibration can simultaneously excite the rotor to precess at the torsional frequency or its multiples, and to precess at the opposite frequency. When the torsional vibration is symmetrical about the equilibrium position, the amplitudes of the excited precession and the opposite precession are similar, and the rotor shaft trajectory is close to a straight line or a very flat ellipse.

[0082] It is understandable that the physical mechanism of instability is as follows: when the above frequency conditions are met, torsional vibration continuously converts its own energy into rotor bending vibration energy through the gyro effect, forming an equivalent positive feedback channel, which causes bending vibration energy to accumulate continuously.

[0083] It should also be noted that whether instability occurs depends on the amplitude of the torsional vibration and the magnitude of the static displacement. If the amplitude of the torsional vibration is large, the input excitation energy is greater than the energy dissipated by the damping of the rotor system, and the rotor's bending self-excited vibration will cause instability.

[0084] If the rotor's torsional vibration is generated by fluid excitation, the torsional vibration continuously obtains energy from the fluid and continuously converts it into rotor bending vibration energy. The aforementioned instability conditions can be figuratively viewed as the switching conditions of an energy conversion channel. When these conditions are met, the switch is open, energy is input, and the rotor becomes unstable. Whether it is forward or reverse precession instability, it is asynchronous vibration, which will generate alternating bending stress on the shaft.

[0085] It should also be noted that the conditions for positive precession instability include: This can reflect the coupling between torsional vibration and anti-precession modes, which may induce forward precession instability; the conditions for anti-precession instability include This can reflect the coupling between torsional vibration and forward precession mode, which may trigger reverse precession instability.

[0086] In practical applications, determining whether a rotor system is actually unstable requires two steps: (a) checking whether the above frequency conditions are met; and (b) assessing whether the amplitude of the torsional vibration is large enough under these conditions to allow the input energy to overcome the system's damping. Both conditions must be met simultaneously.

[0087] Step 105: Based on the instability criterion, perform a stability assessment on the rotor system, predict the dangerous operating conditions that may cause bending self-excited vibration, and determine the corresponding avoidance measures based on the assessment results.

[0088] Understandably, this step applies the aforementioned theoretical criteria to engineering practice.

[0089] In one possible embodiment, based on the instability criterion, the forward precession natural frequency and the reverse precession natural frequency of the rotor system at the operating speed are calculated, including: Positive precession natural frequency The characteristic equation is: ; Reverse precession natural frequency The characteristic equation is: ; in, For quality, The moment of inertia of the diameter. The moment of inertia is the polar rotation. The average rotational speed, , , , This is the stiffness coefficient; This represents the normal precession frequency. It represents the reverse precession natural frequency.

[0090] like Figure 3 As shown, for a given rotor system (such as the low-pressure rotor of a certain type of aero-engine), calculate its operating speeds. The forward and reverse precession natural frequencies and Obtain the possible torsional vibration characteristics (fundamental frequency) of the rotor. and the amplitude of each harmonic). , and By comparing the results and identifying all potential instability points (combinations of dangerous speed and torsional harmonic order) based on the instability criteria, a comparison is made.

[0091] The low-pressure rotor of a certain dual-rotor engine, if calculated at the cruising speed... The torsional vibration fundamental frequency caused by combustion oscillation Then according to the criterion The system is at risk of first-order torsional vibration excitation and positive precession instability during cruise.

[0092] At the same time, it can draw stability domain diagrams with rotational speed and torsional vibration frequency as parameters, intuitively identifying dangerous operating conditions.

[0093] In one possible embodiment, the preventive measures include at least one of the following: adjusting the structural parameters of the rotor system so that its forward precession natural frequency and reverse precession natural frequency avoid the dangerous frequency conditions determined by the instability criterion; avoiding torsional resonance of the rotor by design; adding a damper to the rotor system; and for rotors with toothed connections, ensuring that the cylindrical centering surfaces at both ends of the connection are tightly fitted to reduce static displacement.

[0094] It is worth noting that torsional vibration simultaneously excites the rotor to precess at the torsional frequency or its harmonics, both forward and reverse. When the torsional vibration is symmetrical about the equilibrium position, the amplitudes of the excited forward and reverse precessions are close, and the rotor shaft's trajectory is close to a straight line or a very flat ellipse. During shaft stress testing, if the sampling frequency is too low or improperly set, the maximum alternating stress may not be measured.

[0095] For example, based on the assessment results, the following targeted preventative measures are proposed to avoid or reduce the risk of instability: Frequency avoidance: Adjust the support stiffness and mass distribution of the rotor system to change its forward / reverse precession natural frequency, so that it avoids all dangerous combinations of conditions within its operating range.

[0096] Suppressing torsional vibration: Optimize the design to avoid torsional resonance in the rotor and reduce the amplitude of torsional vibration at the source.

[0097] Increase damping: Install dampers (such as squeeze film dampers) in the rotor system to improve the system's ability to dissipate energy, and avoid actual instability even if the frequency conditions are partially met.

[0098] Reduce static displacement: For rotors using geared connections (such as low-pressure turbine shaft and fan shaft), ensure that the cylindrical centering surfaces at both ends of the connection are tightly fitted to avoid increased static displacement due to increased rotor flexibility caused by increased clearance, thereby reducing the risk of instability.

[0099] Monitoring and Alarm: In the airborne vibration monitoring system, real-time monitoring of asynchronous vibration components is enhanced, and vibration thresholds calculated based on this method are set for early warning.

[0100] In summary, this invention, by establishing an accurate periodic time-varying model, introducing static displacement, and employing an improved perturbation-harmonic balance method for analytical solution, ultimately obtains a set of instability criteria with clear physical meaning and complete form. This method can not only be used for the stability design and verification of rotor systems but also provide a theoretical basis for fault diagnosis and health management. In particular, this method reveals that the trajectory of such self-excited vibrations may be a very flattened ellipse or an approximately straight line, suggesting that a sufficiently high sampling frequency is required in experimental testing to accurately capture the maximum alternating stress, which has significant engineering guiding value.

[0101] like Figure 4 As shown, Figure 4 This is a flowchart illustrating a calculation method for torsional vibration-induced rotor bending self-excited vibration proposed in an embodiment of this application. First, considering multi-order harmonic fluid torsional excitation, a time-varying differential Hill equation for the rotor under torsional vibration conditions is established. Then, using the axis without self-weight as the origin of the coordinate system, a vibration equation for the rotor system including gravity terms is established. The perturbation method is used to solve this equation. It is determined whether the system exhibits a response that increases with time, thereby establishing the criterion for rotor bending vibration instability. The forward and reverse precession natural frequencies of the given rotor system are calculated, and it is evaluated whether each order of torsional vibration harmonic components satisfies the instability condition, predicting the dangerous bending self-excited vibration condition of the system. Finally, mitigation measures are proposed to address the instability risk.

[0102] This application proposes an analytical method for torsional vibration-induced bending self-excited vibration of a rotor. The method establishes a periodic time-varying vibration equation for the rotor system. Static displacement terms generated by gravity, inertial force, and inertial torque are introduced into the periodic time-varying vibration equation to construct a periodic time-varying vibration equation containing static displacement terms. Based on the perturbation method and harmonic balance method, the periodic time-varying vibration equation containing static displacement terms is solved to obtain the multi-order approximate response of the rotor system under torsional vibration excitation. Based on whether there are time-increasing terms in each order of approximate response, an instability criterion for bending self-excited vibration of the rotor system is determined. Based on the instability criterion, the rotor system is analyzed... The system performs stability assessments to predict dangerous conditions that may lead to bending self-excited vibrations, and determines corresponding avoidance measures based on the assessment results. Since traditional vibration analysis linearly decouples bending and torsional vibrations, resulting in large errors under high speeds and strong excitation, this scheme establishes a periodic time-varying vibration equation (Hill equation) that incorporates gyroscopic effects and bending-torsional coupling. This allows for a complete mathematical modeling of the physical quantities that cause the system stiffness to vary due to the gyroscopic effect during torsional vibration, overcoming the shortcomings of traditional linearization methods under strong coupling conditions. Furthermore, by introducing static displacement terms generated by gravity, inertial force, and inertial torque into the periodic time-varying vibration equation,... This method explicitly considers the contribution of static displacement caused by gravity and inertial forces during maneuvering to the bending-torsional coupling effect, thus significantly improving stability and computational accuracy. Then, it employs a perturbation method with torsional amplitude as a small parameter, combined with a harmonic balance method for step-by-step analytical solutions. Compared to purely numerical methods (such as the Flokai method), this approach gradually reveals how torsional vibration energy couples with static displacement and natural frequency through various harmonics and is transferred to bending vibration. It fundamentally elucidates the intrinsic physical meaning of torsional vibration exciting self-excited bending vibration, rather than merely providing numerical conclusions. Furthermore, based on the analytical solution process, this method can systematically... This method derives complete and clear instability criteria and provides targeted preventive measures. Based on this, the solution establishes an accurate time-varying model, considers key static displacement effects, adopts a mechanism-transparent analytical solution, derives systematic instability criteria, and forms a complete evaluation process. This solves the problem of inaccurate predictions caused by neglecting the time-varying nature of gyroscopic torque, unclear physical mechanisms, and lack of systematic criteria when analyzing rotor bending self-excited vibration induced by torsional vibration. This method combines theoretical rigor with engineering practicality and is particularly suitable for the bending-torsional coupled dynamic design and stability analysis of high-end rotating machinery such as aero-engines.

[0103] Compared with the prior art, the present invention has the following significant advantages: 1. At the level of mechanism revelation, the analytical perturbation method employed in this invention possesses high transparency, clearly elucidating the complete physical picture of torsional vibration-induced rotor bending self-excited vibration. Its core mechanism lies in the fact that torsional vibration, through the gyroscopic effect, forms an energy channel, continuously converting its own energy into the energy of rotor bending vibration. Furthermore, the static displacement caused by gravity plays a crucial coupling role in this energy transfer process. Further quantitative analysis reveals the correspondence between the self-excited vibration frequency and the torsional vibration frequency, as well as the direct influence of torsional vibration amplitude and rotor flexibility (static displacement) on the intensity of vibration. The larger the torsional vibration amplitude, the more intense the self-excited vibration; the larger the static displacement, i.e., the greater the rotor flexibility, the greater the self-excited vibration, thus making the intrinsic cause of the instability phenomenon readily apparent.

[0104] 2. This method considers the effect of gravity: This invention overcomes the limitation of traditional analysis that ignores static deformation by explicitly taking into account the static displacement effect caused by gravity (and the inertial force of maneuvering flight). This improvement not only makes the model more in line with engineering practice, but also directly derives a conclusion with important guiding value: the greater the rotor flexibility (e.g., a low-pressure rotor with a slender shaft), the more prone it is to instability under torsional vibration excitation. This provides a clear direction for the design optimization of key components.

[0105] 3. In terms of practical output, this invention systematically derives a complete and quantitative set of instability criteria. These criteria comprehensively cover all harmonics of torsional vibration (…). The study identifies all possible frequency conditions for positive and negative precession instability, resolving the issues of incomplete criteria and ambiguous physical meaning in existing methods, and providing a direct and reliable theoretical basis for stability assessment.

[0106] 4. This method can identify elliptical trajectories: When torsional vibration is symmetrical, the excited bending self-excited vibration trajectory is a very flat ellipse or an approximately straight line. This discovery provides important theoretical clues for experimental verification and fault diagnosis, suggesting that a sufficiently high sampling frequency must be used in the test to accurately capture the maximum alternating stress, avoiding the risk of missed detection due to improper monitoring settings.

[0107] 5. Finally, all the above theoretical results have direct engineering application value. This method can be directly used to guide the bending-torsional coupling dynamic design of rotating machinery rotor systems and to formulate design criteria to prevent such instability. At the same time, the derived criteria and thresholds can also serve as the theoretical basis for setting early warning parameters in airborne vibration monitoring systems, thereby realizing risk prevention and control throughout the entire process from design to operation monitoring.

[0108] The steps described above are for clarity only. In implementation, they can be combined into one step, or some steps can be broken down into multiple steps, as long as they involve the same logical relationship, they are all within the scope of protection of this application. Adding insignificant modifications or introducing insignificant designs to the algorithm or process, without changing the core design of the algorithm and process, are also within the scope of protection of this application.

[0109] Another embodiment of this application proposes an analysis system for torsional vibration-induced rotor bending self-excited vibration. The details of this analysis system are described below. The following content is for illustrative purposes only and is not essential for implementing this example. Figure 5 This is a schematic diagram of the structure of an analysis system for torsional vibration-excited rotor bending self-excited vibration proposed in this embodiment, including: Equation building module 210 is used to establish the periodic time-varying vibration equation of the rotor system; The static displacement processing module 220 is used to introduce static displacement terms generated by gravity, inertial force and inertial torque into the periodic time-varying vibration equation to construct a periodic time-varying vibration equation containing static displacement terms. The vibration response solution module 230 is used to solve the periodic time-varying vibration equation containing static displacement terms based on the perturbation method and the harmonic balance method, so as to obtain the multi-order approximate response of the rotor system under torsional vibration excitation. The instability criterion determination module 240 is used to determine the instability criterion for the rotor system to undergo bending self-excited vibration based on whether there is a term that increases with time in each order of approximate response. The stability assessment and decision module 250 is used to assess the stability of the rotor system based on the instability criterion, predict dangerous operating conditions that may cause bending self-excited vibration, and determine corresponding avoidance measures based on the assessment results.

[0110] It is not difficult to see that this embodiment is a system embodiment corresponding to the above method embodiments, and this embodiment can be implemented in conjunction with the above method embodiments. The relevant technical details and technical effects mentioned in the above method embodiments are still valid in this embodiment, and will not be repeated here to reduce repetition. Accordingly, the relevant technical details mentioned in this embodiment can also be applied to the above method embodiments.

[0111] It is worth mentioning that all modules and units involved in this embodiment are logical modules. In practical applications, a logical unit can be a physical unit, a part of a physical unit, or a combination of multiple physical units. Furthermore, to highlight the innovative aspects of this application, this embodiment does not introduce units that are not closely related to solving the technical problems proposed in this application; however, this does not mean that other units do not exist in this embodiment.

[0112] Another embodiment of this application provides an electronic device, such as Figure 6 As shown, it includes a processor 31 and a memory 32. The memory 32 stores instructions that the processor 31 can execute. When the processor 31 is configured to execute the instructions, the electronic device can realize an analysis method for torsional vibration-induced rotor bending self-excited vibration as described in the above method embodiment.

[0113] The memory and processor are connected via a bus, which includes any number of interconnecting buses and bridges, connecting various circuits of one or more processors and the memory. The bus can also connect various other circuits such as peripheral devices, voltage regulators, and power management circuits, which are well known in the art and will not be described further herein. The bus interface provides an interface between the bus and the transceiver. The transceiver can be a single component or multiple components, such as multiple receivers and transmitters, providing a unit for communicating with various other devices over a transmission medium. Data processed by the processor is transmitted over the wireless medium via an antenna, which further receives data and transmits it to the processor.

[0114] The processor manages the bus and general processing, and also provides various functions, including timing, peripheral interfaces, voltage regulation, power management, and other control functions. Memory is used to store data used by the processor during operation.

[0115] Another embodiment of this application proposes a computer-readable storage medium storing a computer program that, when executed by a processor, can implement an analysis method for torsional vibration-induced rotor bending self-excited vibration as described in the above method embodiments.

[0116] That is, those skilled in the art will understand that all or part of the steps in the above method embodiments can be implemented by a program instructing related hardware. The program is stored in a storage medium and includes several instructions to cause a device (such as a microcontroller, chip, etc.) or processor to execute all or part of the steps of the method described in the method embodiments of this application. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory, random access memory, magnetic disks, or optical disks.

[0117] Those skilled in the art will understand that the above embodiments are specific implementations of this application, and in practical applications, various changes can be made in form and detail without departing from the spirit and scope of this application. For those skilled in the art, several improvements and modifications can be made without departing from the principles of this application, and these improvements and modifications are also considered to be within the scope of protection of this application.

Claims

1. An analytical method for analyzing torsional vibration-induced bending self-excited vibration of a rotor, characterized in that, include: The periodic time-varying vibration equation of the rotor system is established, including: establishing the vibration equation of the rotor system that includes gyroscopic effects and bending-torsional coupling, and defining the total transient rotation angle of the rotor as the sum of the average rotational speed and the torsional vibration component; expanding the torsional vibration component into a complex Fourier series containing Nth harmonics; substituting the torsional vibration in the form of the complex Fourier series into the rotor system vibration equation to obtain the periodic time-varying vibration equation. In the periodic time-varying vibration equation, a static displacement term generated by gravity, inertial force, and inertial torque is introduced to construct a periodic time-varying vibration equation that includes a static displacement term. This includes: setting the origin of the coordinate system of the periodic time-varying vibration equation from the geometric axis to the static equilibrium position of the rotor when it does not undergo torsional vibration under the combined action of gravity, inertial force, and inertial torque; and introducing a gravity term on the right-hand side of the periodic time-varying vibration equation to consider the static displacement caused by gravity. Based on the perturbation method and harmonic balance method, the periodic time-varying vibration equation containing static displacement terms is solved to obtain the approximate responses of the rotor system under torsional vibration excitation. This includes: expanding the solution of the periodic time-varying vibration equation containing static displacement terms into a perturbation series with the amplitude of the rotor torsional vibration as a small parameter; solving the zero-order approximate response of the perturbation series to obtain the asynchronous bending vibration response containing static displacement caused by gravity and inertial forces, as well as vibrations at the rotor's forward and reverse precession natural frequencies; substituting the zero-order approximate response into the periodic time-varying vibration equation to solve the first-order approximate equation to obtain the response generated by the combined excitation of torsional vibration harmonics and natural frequencies; substituting the zero-order and first-order approximate responses into the periodic time-varying vibration equation to construct and solve the second-order approximate equation, and applying the harmonic balance method to solve it to obtain the second-order approximate response and further verify the instability condition. Based on whether there are terms that increase with time in each order of approximate response, the instability criterion for the rotor system to undergo bending self-excited vibration is determined; Based on the instability criterion, the stability of the rotor system is assessed to predict the dangerous operating conditions that may cause bending self-excited vibration, and based on the assessment results, corresponding avoidance measures are determined.

2. The analytical method for analyzing torsional vibration-induced rotor bending self-excited vibration according to claim 1, characterized in that, Instability criteria for rotor systems experiencing bending self-excited vibrations include: forward precession instability criteria and reverse precession instability criteria; The instability criterion for determining the bending self-excited vibration of the rotor system based on whether there is a term that increases with time in each order of approximate response includes: When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experienced bending forward precession instability; When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experienced bending reverse precession self-excited vibration instability; When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experienced bending forward precession instability; When the rotor torsional vibration First harmonic frequency satisfy Or, the fundamental frequency of torsional vibration. satisfy At that time, the rotor system experienced bending reverse precession self-excited vibration instability; in, For harmonic orders, ; It is the positive precession natural frequency. It is the natural frequency of the reverse precession.

3. The analytical method for torsional vibration-induced rotor bending self-excited vibration according to claim 2, characterized in that, The stability assessment of the rotor system includes: Based on the instability criterion, the forward precession natural frequency and the reverse precession natural frequency of the rotor system at the operating speed are calculated. By comparing the positive and negative precession frequencies with the instability criterion, the torsional vibration harmonic order that causes instability and its corresponding critical speed are determined. Based on the torsional vibration harmonic order that triggers instability and its corresponding critical speed, and combined with the actual torsional vibration amplitude and damping parameters of the rotor system, it is determined whether instability has occurred.

4. The analytical method for torsional vibration-induced rotor bending self-excited vibration according to claim 2, characterized in that, The calculation of the forward precession natural frequency and reverse precession natural frequency of the rotor system at the operating speed based on the instability criterion includes: Positive precession natural frequency The characteristic equation is: ； Reverse precession natural frequency The characteristic equation is: ； in, For quality, The moment of inertia of the diameter. The moment of inertia is the polar rotation. The average rotational speed, , , , This is the stiffness coefficient; This represents the natural frequency of positive precession. It represents the reverse precession natural frequency.

5. The analytical method for torsional vibration-induced rotor bending self-excited vibration according to claim 1, characterized in that, Preventive measures include at least one of the following: Adjust the structural parameters of the rotor system so that its forward and reverse precession natural frequencies avoid the dangerous frequency conditions determined by the instability criterion; Torsional resonance in the rotor is avoided through design; Add a damper to the rotor system; For rotors with toothed connections, ensure that the cylindrical centering surfaces at both ends of the connection are in close fit to reduce static displacement.

6. An electronic device, characterized in that, include: A processor and a memory, wherein the memory stores instructions that the processor can execute, and the processor is configured to, when executing the instructions, enable the electronic device to implement an analysis method for torsional vibration-induced rotor bending self-excited vibration as described in any one of claims 1 to 5.

7. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it can implement an analysis method for torsional vibration-induced rotor bending self-excited vibration as described in any one of claims 1 to 5.

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