Rotating machinery self-excited vibration phase tuning prediction avoidance method based on symmetry principle
By using a self-excited vibration prediction method based on the principle of symmetry, the problem of predicting self-excited vibration of rotating machinery is solved, and vibration suppression and parameter optimization of rotating machinery are achieved. This method is applicable to a variety of rotating machinery devices.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- TIANJIN UNIV
- Filing Date
- 2026-03-03
- Publication Date
- 2026-07-10
AI Technical Summary
Existing technologies make it difficult to predict the conditions under which self-excited vibrations of rotating machinery will occur in advance, which can lead to increased vibration amplitude and potentially cause mechanical damage. It is necessary to provide a mapping relationship for self-excited vibrations so that resonance can be suppressed through parameter selection.
Based on the principle of symmetry, by calculating the spatiotemporal excitation phase and Fourier series characterization of rotating machinery, the self-excited vibration mode of rotating machinery is analyzed using the superposition principle, and the conditions for the occurrence of self-excited vibration are predicted.
It provides an accurate, universal, and efficient method for predicting self-excited vibrations, supports parameter selection during the design phase, reduces costs, and is applicable to a variety of rotating machinery devices.
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Figure CN122365736A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of predicting the dynamic characteristics of rotating machinery, and in particular to a method for predicting and avoiding the phase tuning of self-excited vibrations of rotating machinery based on the principle of symmetry. Background Technology
[0002] To achieve energy conversion or the transmission of motion and power, various rotating machines are widely used in engineering, such as rotating electric machines, turbines, planetary drives, rolling bearings, piston motors, and gear drives. These machines typically consist of two or more central components and intermediate force-transmitting components. The former include components such as the stator, rotor, sun gear, planetary carrier, ring gear, wave generator, internal cam, distribution shaft, and cylinder; the latter includes components such as rotor poles, stator teeth, planetary gears, rolling elements, gears, and pistons. To achieve the basic functions of rotating machinery, there must be relative motion between the central components, and these components can generate time-varying forces with the intermediate force-transmitting components, leading to vibration, noise, and accuracy issues.
[0003] In order to suppress vibration noise and improve operating accuracy, existing technologies have proposed a variety of effective solutions. Among them, parameter modification is a common method to suppress or even eliminate time-varying excitation by changing the matching of basic parameters, thereby achieving performance improvement. This is a low-cost method that can be implemented in the design stage. However, the prerequisite for implementing this method is that the accurate mapping relationship between basic parameters and time-varying internal excitation and dynamic behavior must be obtained. Among the various information describing time-varying excitation, the excitation phase is a measure closely related to the symmetry of rotating machinery, and it is also a bridge connecting basic parameters and dynamic behavior. In the field of planetary gear transmission, reference [1] studied the typical force mode and vibration mode of the central component of planetary gear transmission based on the rigid body assumption, and revealed the phase tuning law of rigid body vibration. Reference [2] explored the typical rigid-flexible coupling vibration behavior of planetary transmission gear ring based on the flexible body assumption, and obtained the corresponding phase tuning law. Reference [3] studied the mapping relationship between the number of ripples and rolling elements of the inner and outer raceways of the bearing and the three rigid body vibration modes of the bearing, and also compared it with previous literature. Reference [4] abandons the traditional odd-even plunger classification research method and clarifies the influence relationship between the number of cam actions and the number of plungers and the typical rigid body vibration mode by introducing the plunger phase. Based on the existing research, Reference [5] gives the spatiotemporal phase relationship of time-varying internal excitation of common rotating machinery and accurately describes the leading, lagging or in-phase relationship of time-varying forces at each intermediate component.
[0004] In fact, time-varying excitation can induce not only forced vibration but also self-excited vibration. This vibration originates from the elastic deformation of the central component. In this case, the time-varying excitation can generate a time-varying axial force, thereby aggravating the vibration amplitude of the central component. The increased vibration amplitude further leads to a larger axial force, resulting in more severe axial vibration. When the excitation phase and the elastic vibration wavenumber of the central component meet certain conditions, severe vibration can occur, leading to instability and even serious damage. Self-excited vibration is usually a harmful vibration that seriously affects the dynamic performance of rotating machinery and must be suppressed by appropriate measures. If the conditions for the occurrence of self-excited vibration can be predicted in advance, especially by providing the mapping relationship between basic parameters and possible self-excited resonances, it will be very beneficial to achieve vibration reduction, noise reduction, and accuracy improvement through parameter selection.
[0005] To reveal the laws governing the self-excited vibration of rotating machinery, clarify the mapping relationship between basic parameters and self-excited vibration behavior, and propose practical techniques to suppress such resonances, this invention takes a three-center rotating machinery as a typical example in the engineering field and presents a phase tuning prediction and resonance avoidance technique for the self-excited vibration of rotating machinery based on the principle of symmetry.
[0006] References
[0007] [1]PARKER R G. A physical explanation for the effectiveness of planetphasing to suppress planetary gear vibration [J]. Journal of Sound andVibration, 2000, 236(4): 561–573.
[0008] [2]WANG SY, HUO MH, ZHANG C, et al. Effect of mesh phase on wavevibration of spur planetary ring gear [J]. European Journal of Mechanics-A / Solids, 2011, 30: 820–827.
[0009] [3] Wang Shiyu, Huo Mina, Chen Dongliang, et al. Methods for improving the performance of rolling bearings [P]. Chinese Invention Patent, Authorization No.: ZL201010562863.X.
[0010] [4] Wang Shiyu, Huo Mina, Chen Dongliang, et al. A method for improving the performance of multi-acting internal curve radial piston pumps or motors [P]. Chinese Invention Patent, Authorization No.: ZL201110093939.3.
[0011] [5]WANG SY, MEESAP C. Investigation on mesh and sideband vibrations of a helical planetary ring gear using structure, excitation and deformationsymmetries [J]. Chinese Journal of Mechanical Engineering, 2018, 31: 104. Summary of the Invention
[0012] This invention provides a phase tuning prediction and avoidance method for self-excited vibration of rotating machinery based on the principle of symmetry. This invention provides technical support for time-varying self-excited force cancellation, vibration behavior prediction, and parameter selection of rotating machinery. This technology has significant advantages such as accuracy, universality, and high efficiency, as detailed below:
[0013] A method for predicting and avoiding the phase tuning of self-excited vibrations in rotating machinery based on the principle of symmetry, the method comprising:
[0014] Rotating machinery can be viewed as a rotating mechanism consisting of several coaxially mounted central components and several intermediate force-transmitting components mounted circumferentially.
[0015] The spatiotemporal excitation phase of each intermediate component is calculated based on the spatial rotational symmetry of the topological structure and the temporal translational symmetry of the time-varying excitation.
[0016] Based on symmetry, Fourier series characterization of time-varying excitation at each intermediate component is given; considering the elastic out-of-plane vibration of the central component, the out-of-plane component of the time-varying excitation is calculated, and the resultant force and resultant moment in the six degrees of freedom are calculated using the superposition principle.
[0017] By utilizing the properties of trigonometric functions to analyze the characteristics of the resultant force and resultant moment, the typical self-excited vibration modes of the central component can be predicted.
[0018] The time-varying excitation is: The time-varying excitation at the i-th intermediate component is expressed using a Fourier series:
[0019]
[0020] In the formula, l represents the harmonic order of the time-varying excitation. and Let l be the l-th harmonic coefficient at the i-th intermediate component. Let be the time excitation phase at the i-th intermediate component. This represents the transverse amplitude.
[0021] The self-excited force of the inner central component is:
[0022]
[0023] The in-plane component of the excitation force acting on the inner central member is:
[0024]
[0025] Decompose the in-plane force along the tangential and radial directions to obtain:
[0026]
[0027] Where A0 is the maximum lateral vibration displacement at the edge of the component, and m is the number of nodal diameters. For time-varying excitation frequency, The tangential component of the meshing force; The radial component of the meshing force. For gear pressure angle, It is an integer. The rotational speed of the inner center component is The rotational speed of the central component is , Let be the circumferential position angle of the i-th intermediate component. The angle between the meshing force caused by lateral vibration and the axial direction.
[0028] The resultant force of the time-varying excitation acting on the inner central component is expressed in the follower system as:
[0029]
[0030] in, The resultant force in the horizontal direction exerted by the planetary gears on the central component under the servo system; The resultant force is in the vertical direction. Let be the horizontal component of the force exerted on the central component by the i-th planetary gear in the follower system; The vertical component of the force.
[0031] Taking the lth harmonic of each harmonic, we can obtain:
[0032]
[0033]
[0034] The resultant moment acting on the inner central member is:
[0035]
[0036] In the formula, The lever arm of the time-varying excitation of the inner central component;
[0037] Taking its lth harmonic, we can obtain:
[0038]
[0039] In the formula, The number of functional units on the central component;
[0040] The total axial force on the inner central member is:
[0041]
[0042] Taking its l-th harmonic, we can obtain:
[0043]
[0044] In the formula, , The torsional moment acting on the inner central component is:
[0045]
[0046] Taking the lth harmonic of each harmonic, we can obtain:
[0047]
[0048]
[0049] In the formula, , , , .
[0050] The beneficial effects of the technical solution provided by this invention are:
[0051] 1. This invention fully utilizes the geometric structure and symmetry of rotational motion of rotating machinery, and adopts a superposition method to provide a vibration analysis method for a class of three-center rotating machinery. This method is based on a model-free analysis approach and has the advantages of accuracy, efficiency and simplicity, providing an effective prediction tool for the self-excited vibration response of rotating machinery.
[0052] 2. The self-excited vibration prediction method proposed in this invention can design the selection principles of basic parameters for the application scenario during the design stage, which has the significant advantage of low cost.
[0053] 3. The self-excited vibration prediction method proposed in this invention has the advantage of universality and can be used for prediction and analysis of the self-excited vibration behavior of three-center component transmission devices such as planetary gear transmission, star gear transmission, permanent magnet planetary transmission, live gear transmission, harmonic transmission, rolling bearing and radial piston motor.
[0054] 4. The self-excited vibration prediction method proposed in this invention has the significant advantage of easy expansion. Based on this method, a self-excited vibration law prediction method for two-center components and multi-stage transmission devices can be further invented. Attached Figure Description
[0055] Figure 1 A schematic diagram of the three-center component rotation mechanism provided by the present invention;
[0056] Figure 2 This is a schematic diagram of a planetary gear transmission provided by the present invention;
[0057] Figure 3 A schematic diagram of the axial self-excited force of the central component provided by the present invention;
[0058] Figure 4 This is a schematic diagram of gear meshing force decomposition provided by the present invention;
[0059] Figure 5 This is a schematic diagram of the decomposition of the self-excited force of the planetary transmission provided by the present invention. Detailed Implementation
[0060] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be described in further detail below.
[0061] To overcome the technical difficulties in the background art, this invention proposes a prediction and analysis technique for rigid-elastic coupled vibration of rotating machinery based on the modulation principle.
[0062] The rotating machinery is considered as a rotating mechanism consisting of several coaxially mounted central components and several intermediate force-transmitting components mounted circumferentially. Based on the spatial rotational symmetry of the topological structure and the time-translational symmetry of the time-varying excitation, the spatiotemporal excitation phase of each intermediate component is calculated. The Fourier series characterization of the time-varying excitation at each intermediate component is given based on symmetry. Considering the elastic out-of-plane vibration of the central component, the out-of-plane components of the time-varying excitation are calculated, and the resultant force and resultant moment in the six degrees of freedom are calculated using the superposition principle. The characteristics of the resultant force and resultant moment are analyzed using the properties of trigonometric functions, and the typical self-excited vibration modes of the central component are predicted. Details are as follows:
[0063] Rotating machinery can be equated to a rotating mechanism consisting of several coaxially mounted central components and several intermediate components mounted circumferentially, such as... Figure 1As shown in the figure, this illustration only uses a rotating machine with 3 central components and N intermediate components as an example, but in fact, the technology of this embodiment can be applied to rotating machines with two or more central components.
[0064] Taking the inner central member as the research object, its lateral vibration displacement in the member's follower coordinate system can be expressed as:
[0065] (1)
[0066] In the formula, m is the number of nodal diameters. The position angle, For time-varying excitation frequency, Let be the amplitude of the mid-surface vibration at radius r, which is only a function of r.
[0067] like Figure 3 As shown, for the case of small amplitude vibration, we have:
[0068] (2)
[0069] in, The angle between the meshing force caused by lateral vibration and the axial direction; This represents the lateral vibration displacement.
[0070] The self-excited force acting on the inner central member can be expressed as:
[0071] (3)
[0072] In the formula, A0 is the maximum lateral vibration displacement at the edge of the component. The amplitude is the transverse amplitude. The time-varying excitation at the i-th (i=1, 2, 3, … N) intermediate component can be given a general expression using Fourier series:
[0073] (4)
[0074] In the formula, l represents the harmonic order of the time-varying excitation. and Let l be the l-th harmonic coefficient at the i-th intermediate component. Let be the time excitation phase at the i-th intermediate component, and:
[0075] (5)
[0076] In the formula, The number of functional units on the central component. Let be the circumferential position angle (spatial phase) of the i-th intermediate component, and .
[0077] Assuming the outer central member is fixed, the inner central member is the driving member, and the middle central member is the driven member, the rotational speed of the inner central member is... The rotational speed of the central component is Then the circumferential position angle of the excitation point ,in and The value should make satisfy Z represents the set of integers. It is an integer. Take... ,but Substituting it into equation (3) yields:
[0078] (6)
[0079] The in-plane component of the excitation force acting on the inner central member is:
[0080] (7)
[0081] like Figure 4 As shown, by decomposing the in-plane component of equation (7) along the tangential and radial directions, we can obtain:
[0082] (8)
[0083] in, The tangential component of the meshing force; The radial component of the meshing force. This is the gear pressure angle, typically 20°.
[0084] Projecting the aforementioned radial and tangential forces onto the servo system, we obtain:
[0085] (9)
[0086] in, Let be the horizontal component of the force exerted on the central component by the i-th planetary gear in the follower system; The vertical component of the force.
[0087] According to equations (8) and (9), the resultant force of the time-varying excitation acting on the inner central component can be expressed in the follower system as:
[0088] (10)
[0089] in, The resultant force in the horizontal direction exerted by the planetary gears on the central component under the servo system; The resultant force is in the vertical direction.
[0090] Substituting the lth harmonic of the meshing force in equation (4) into equation (10), we get:
[0091] (11)
[0092] In the formula
[0093] , , ,
[0094] For integers n and N, according to the properties of trigonometric functions, we have:
[0095] (12)
[0096] According to equation (12), equation (11) can be simplified to:
[0097] (13)
[0098] Similarly, we can conclude that:
[0099] (14)
[0100] in, The l-th harmonic component of the resultant force in the horizontal direction on the central component; It represents the l-th harmonic component of the resultant force in the vertical direction.
[0101] The resultant moment acting on the inner central member is:
[0102] (15)
[0103] In the formula, The lever arm is the force arm of the time-varying excitation of the inner central component.
[0104] From equations (8) and (15), we can obtain:
[0105] (16)
[0106] Substituting the lth harmonic of the force in equation (4) into equation (16) and simplifying it in conjunction with equation (12), we get:
[0107] (17)
[0108] The total axial force on the inner central member is:
[0109] (18)
[0110] Substituting the lth harmonic of the meshing force in equation (4) into equation (18) and simplifying it in conjunction with equation (12), we get:
[0111] (19)
[0112] In the formula
[0113] ,
[0114] in, It is the l-th harmonic component of the axial resultant force on the central component.
[0115] The torsional moment acting on the inner central component is:
[0116] (20)
[0117] Substituting the lth harmonic of the meshing force in equation (4) into equation (20) and simplifying it in conjunction with equation (12), we get:
[0118] (twenty one)
[0119] Similarly, we can conclude that:
[0120] (twenty two)
[0121] In the formula
[0122] ,
[0123] ,
[0124] in, and The l-th harmonic component of the torsional moment acting on the central component.
[0125] The force between the inner central component and the intermediate force transmission component usually consists of a time-varying component and a steady component. According to the general expression given by Fourier series, when the harmonic order l is 0, it is a steady force. First, considering only the self-excited vibration law caused by the steady component, from equations (12), (13), and (14), we can see that the time-varying excitation component acting on the central component at this time... and The constant value is zero, meaning that the steady force will not cause translational vibration; from equations (12) and (17), it can be seen that the torque acting on the central component at this time is zero. It is never zero and can induce torsional vibration.
[0126] From equation (12) and equation (19), it can be seen that when At that time, the axial self-excited force on the central component is zero. At this time, the axial self-excited force on the central component is not zero, that is, axial translational vibration is excited at this time; from (12) and equations (21) and (22), it can be seen that when When, the torsional torque acting on the central component is zero, when or At this time, the torsional torque acting on the central component is not zero, that is, torsional vibration is excited at this time.
[0127]
[0128] Analyzing the self-excited vibration caused by time-varying forces, since the axial vibration amplitude of the central component is very small, it can be approximated as... Combining the derivation results of equations (12), (13), and (14), it can be seen that when and At that time, the time-varying excitation component acting on the central component and The lth harmonic components are all zero; when and At this time, the first harmonic component of the time-varying excitation component acting on the central component is not zero, that is, translational vibration is excited at this time. From equations (12) and (17), it can be obtained that when When, the lth harmonic component of the torque acting on the central component is zero; when When the first harmonic of the torque acting on the central component is not zero, torsional vibration is excited. The results are shown in Table 2.
[0129]
[0130] From the derivation results of equations (12) and (19), it can be seen that... and When the first harmonic component of the axial self-excited force on the central component is zero, the first harmonic component of the axial self-excited force is zero. or At this time, the first harmonic of the axial self-excited force on the central component is not zero, that is, at this time, axial translational vibration is excited; from equations (12), (21), and (22), it can be obtained that when When, the lth harmonic component of the torsional moment acting on the central component is zero, when or or or At this time, the first harmonic of the torsional moment acting on the central component is not zero, that is, torsional oscillation is excited at this time. The results are shown in Table 3.
[0131]
[0132] Example 2
[0133] The following is combined with Figures 2-4 The described technique, without altering its fundamental principles, can be further improved in many ways to enhance analytical performance. This will be further illustrated below with reference to the embodiments shown in the accompanying drawings.
[0134] This invention provides a rigid-elastic coupling vibration analysis process for planetary gear transmission based on modulation principles, such as... Figure 2 As shown.
[0135] Assuming the sun gear, planet gears, and ring gear have 32, 13, and 58 teeth respectively, and the planet gears are mounted on the planet carrier, the number of which is... For ease of analysis, a phase tuning factor is defined. , That is According to the phase tuning theory proposed in this embodiment of the invention, the steady part of the meshing force will definitely excite in-plane torsional vibration and suppress translational vibration; when the pitch diameter m is 2, it will suppress axial translational vibration and out-of-plane torsional vibration. For the time-varying part of the meshing force, the laws shown in Tables 4 and 5 can be obtained. When the harmonic order is 1, both in-plane translational and torsional vibrations are suppressed. This parameter selection is suitable for planetary gear transmission scenarios where both in-plane translational and torsional vibrations are highly sensitive. When the harmonic order is 5, both out-of-plane axial translational and torsional vibrations are suppressed. This parameter selection is suitable for planetary gear transmission scenarios where both out-of-plane axial and torsional vibrations are highly sensitive.
[0136]
[0137]
[0138] Those skilled in the art will understand that the accompanying drawings are merely schematic diagrams of a preferred embodiment, and the sequence numbers of the above embodiments of the present invention are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.
[0139] Unless otherwise specified, the model numbers of the various devices in this embodiment of the invention are not limited, and any device that can perform the above functions is acceptable.
[0140] Those skilled in the art will understand that the accompanying drawings are merely schematic diagrams of a preferred embodiment, and the sequence numbers of the above embodiments of the present invention are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.
[0141] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A method for predicting and avoiding the phase tuning of self-excited vibrations in rotating machinery based on the principle of symmetry, characterized in that, The method includes: Rotating machinery can be viewed as a rotating mechanism consisting of several coaxially mounted central components and several intermediate force-transmitting components mounted circumferentially. The spatiotemporal excitation phase of each intermediate component is calculated based on the spatial rotational symmetry of the topological structure and the temporal translational symmetry of the time-varying excitation. Based on symmetry, Fourier series characterization of time-varying excitation at each intermediate component is given; considering the elastic out-of-plane vibration of the central component, the out-of-plane component of the time-varying excitation is calculated, and the resultant force and resultant moment in the six degrees of freedom are calculated using the superposition principle. By utilizing the properties of trigonometric functions to analyze the characteristics of the resultant force and resultant moment, the typical self-excited vibration modes of the central component can be predicted.
2. The method for predicting and avoiding the phase tuning of self-excited vibration in rotating machinery based on the principle of symmetry, as described in claim 1, is characterized in that... The time-varying excitation is: The time-varying excitation at the i-th intermediate component is expressed using a Fourier series: ； In the formula, l represents the harmonic order of the time-varying excitation. and Let l be the l-th harmonic coefficient at the i-th intermediate component. Let be the time excitation phase at the i-th intermediate component. This represents the transverse amplitude.
3. The method for predicting and avoiding the phase tuning of self-excited vibration in rotating machinery based on the principle of symmetry, as described in claim 2, is characterized in that... The self-excited force of the inner central component is: ； The in-plane component of the excitation force acting on the inner central member is: ； Decompose the in-plane force along the tangential and radial directions to obtain: ； Where A0 is the maximum lateral vibration displacement at the edge of the component, and m is the number of nodal diameters. For time-varying excitation frequency, The tangential component of the meshing force; The radial component of the meshing force. For gear pressure angle, It is an integer. The rotational speed of the inner center component is The rotational speed of the central component is , Let be the circumferential position angle of the i-th intermediate component. The angle between the meshing force caused by lateral vibration and the axial direction.
4. The method for predicting and avoiding the phase tuning of self-excited vibration in rotating machinery based on the principle of symmetry, as described in claim 3, is characterized in that... The resultant force of the time-varying excitation acting on the inner central component is expressed in the follower system as: ； in, The resultant force in the horizontal direction exerted by the planetary gears on the central component under the servo system; The resultant force is in the vertical direction. Let be the horizontal component of the force exerted on the central component by the i-th planetary gear in the follower system; The vertical component of the force.
5. The method for predicting and avoiding the phase tuning of self-excited vibration in rotating machinery based on the principle of symmetry, as described in claim 4, is characterized in that... The resultant moment acting on the inner central member is: ； In the formula, The lever arm of the time-varying excitation of the inner central component; The lth harmonic can be obtained as follows: ； In the formula, The number of functional units on the central component; The total axial force on the inner central member is: ； The lth harmonic can be obtained as follows: ； In the formula, , .