Aviation gear resonance prediction and anti-vibration method based on traveling wave resonance theory

Abstract

The invention provides an aviation gear resonance prediction and vibration absorption method based on a traveling wave resonance theory, and relates to the technical field of aviation gear vibration noise. According to the method, based on the gear traveling wave resonance principle and traveling wave resonance characteristics, test analysis is carried out on the aspects of gear structure vibration dynamic characteristics, loading, strength and the like aiming at the phenomenon that the radial plate of the high-speed thin-radial-plate straight gear is fractured, and the vibration characteristics of the thin-radial-plate straight gear during traveling wave resonance are described; a theoretical formula of composite traveling wave resonance is deduced on the basis of experimental data and an existing basic traveling wave resonance theory, a vibration characteristic rule of the composite traveling wave resonance is further revealed by using the formula in a deep level, finally, improvement suggestions are put forward for the structure of the straight gear, and the composite traveling wave resonance formula of the thin radial plate straight gear is verified again.

Description

Technical Field

[0001] This invention relates to the field of aircraft gear vibration and noise technology, and in particular to an aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory. Background Technology

[0002] High-speed thin-spread spur gears are used for commutation and power transmission in the transmission systems of aero-engine accessories. With the continuous iterative development of aero-engines, gear transmission systems operate under harsh conditions such as high speed, heavy load, and high power density, making their structures increasingly complex. This has promoted the application of lightweight thin-walled gears, but their stiffness is weakened, which easily leads to traveling wave vibrations in the system, further causing abnormal system noise. To accurately predict various parameters of aero-engine gear transmission systems under resonance conditions and to implement vibration mitigation, research on the traveling wave resonance theory of aero-engine gear transmission systems urgently needs to be developed.

[0003] Among the many types of microphones, the outgoing noise microphone is a classic type, usually consisting of an acoustic probe, a waveguide, a microphone unit, a preamplifier, and other special structures and housings. The microphone begins to work, which is responsible for converting sound waves (sound pressure changes) into weak electrical signals. Gear resonance has a narrow speed band sensitivity, and when operating at high speeds, with initial defects, and within the dangerous speed range accompanied by traveling wave resonance, gear fatigue fracture will occur quickly. These developments and applications of thin-walled gears will bring two service problems to the transmission system: (1) complex and dense resonances exist under high-speed, heavy-load conditions, including some unpredictable harmful resonances; (2) due to the thin-walled structure and complex operating conditions, the fatigue resistance of thin-walled gears will be reduced. Among the harmful resonances that reduce gear fatigue mentioned above, traveling wave resonance is the most typical and most frequent harmful resonance mode, and it is a recognized challenge faced by thin-walled gears in high-speed transmission systems.

[0004] Currently, scholars both domestically and internationally have only derived expressions for the resonant frequency and resonant speed of a single gear traveling wave, and discussed the excitation principle and resonance conditions of gear traveling wave resonance. However, there are few literature reports on the phenomenon and occurrence conditions of composite traveling wave resonance with two pitch diameter types coexisting. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this method provides a resonance prediction and vibration mitigation method for aircraft gears based on traveling wave resonance theory.

[0006] On the one hand, this invention provides a method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory, comprising the following steps:

[0007] A three-dimensional model of the gear is established and modal analysis is performed. Specifically, the natural frequency of the gear pair under interaction is obtained through static-modal calculation. Then, the natural frequency value of the gear is approximated to the dynamic frequency value to predict the resonant speed of the front and rear traveling waves of the driving spur gear and use it as the boundary condition for transient dynamics calculation. The frequencies of the front and rear traveling waves of the gear are then calculated.

[0008] Step S1: Based on the actual gear geometry parameters and the basic dimensions of the gear shaft, a three-dimensional gear model is established using simulation software. The established three-dimensional gear model is then imported into the pre-processing mesh software HyperMesh. C3D8R hexahedral mesh elements are used, and a two-dimensional mesh mapping method is employed to generate a three-dimensional mesh for mesh generation. Mesh independence tests are then performed on the modal analysis of the active spur gear.

[0009] Step S2: Considering the interaction of the spur gear pair, perform static-modal calculations to obtain the natural frequencies of the gears;

[0010] The specific natural frequency of the gear is:

[0011] In a stationary coordinate system, the expressions for the resonant frequencies of the traveling wave before and after resonance in a gear pitch diameter type are:

[0012] ;

[0013] In the formula The gear meshing frequency, The gear rotation frequency, Including the resonant frequency f of the gear traveling wave F and the resonant frequency f of the traveling wave behind the gear. B m is the gear modal pitch number.

[0014] Step S3: Based on the traveling wave resonance theory, the gear natural frequency value calculated by modal analysis is approximated to the dynamic frequency value. The forward and backward traveling wave resonance speeds of the active spur gear are predicted and used as the boundary conditions for transient dynamics calculation. The forward and backward traveling wave resonance frequencies of the gear are then calculated.

[0015] The resonant rotation speed n F,B Including the traveling wave resonant speed Resonance speed with the back-traveling wave Specifically: ;

[0016] The specific resonant frequency is as follows: In the rotating coordinate system, the excitation force frequency is equal to the resonant frequencies of the preceding and following traveling waves, therefore: ;

[0017] In the formula, k=0.5,1 is the harmonic number of the excitation force, where the amplitude of the excitation force is the largest when k=1, and the natural frequency f of the gear and the frequency f of the excitation force are also related. eWhen they are equal, traveling wave resonance will occur;

[0018] On the other hand, this application proposes an electronic device, including: one or more processors, and a memory for storing instructions, which, when executed by the one or more processors, cause the one or more processors to execute the aforementioned aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory.

[0019] Thirdly, this application proposes a computer-readable storage medium storing executable instructions that, when executed, cause a processor to perform the aforementioned aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory.

[0020] Fourthly, this application proposes a computer program product, including a computer program or instructions, which, when executed by a processor, implements the aforementioned method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory.

[0021] The beneficial effects of adopting the above technical solution are as follows:

[0022] This invention provides a method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory. This method can predict and mitigate the resonance modes of gear transmission systems, thereby improving the service life of aircraft gear transmission systems. Attached Figure Description

[0023] Figure 1 A flowchart illustrating the overall concept of active spur gear prediction for the implementation of this invention;

[0024] Figure 2 A finite element model and mesh generation diagram of the gear shaft provided for the implementation of this invention;

[0025] Figure 3 Mesh independence verification diagram provided for implementation of the present invention;

[0026] Figure 4 The modal displacement distribution diagrams of the first three pitch diameters of the driving spur gear at different frequencies are provided for the implementation of this invention.

[0027] Among them, (a) -1118.9Hz, (b) -1817.9Hz, and (c) -4620.5Hz;

[0028] Figure 5 A schematic diagram showing the location of the acoustic waveguide and the acoustic propagation path for the purpose of implementing this invention;

[0029] Figure 6 A diagram of a noise measurement system provided for the implementation of this invention;

[0030] Figure 7Schematic diagram of radial traveling wave resonance of the two-section active spur gear at different times t for the implementation of this invention;

[0031] Where (a)-t is 0.0161s, (b)-t is 0.0162s, and (c)-t is 0.0206s;

[0032] Figure 8 Schematic diagram of radial traveling wave resonance of the three-section active spur gear at different times t for the implementation of this invention;

[0033] Where (a)-t is 0.0608s, (b)-t is 0.0609s, and (c)-t is 0.0610s;

[0034] Figure 9 A diagram showing the axial vibration displacement at adjacent moments under the traveling wave resonance condition of a three-section active spur gear, provided for the implementation of this invention.

[0035] Figure 10 The traveling wave resonance spectrum diagram of the front-rear travel of the two-pitch diameter of the active spur gear provided for the implementation of the present invention;

[0036] Among them, (a) is the time-frequency domain analysis diagram of the two-section forward traveling wave resonance state, and (b) is the time-frequency domain analysis diagram of the two-section backward traveling wave resonance state.

[0037] Figure 11 The resonance spectrum diagram of the three pitch diameters of the active spur gear provided for the implementation of the present invention;

[0038] Among them, (a) is the time-frequency domain analysis diagram of the forward traveling wave displacement of the three-section diameter, and (b) is the time-frequency domain analysis diagram of the backward traveling wave displacement of the three-section diameter;

[0039] Figure 12 Experimental values ​​of the traveling wave resonant frequency of the three-section diameter active spur gear provided for the implementation of this invention. Detailed Implementation

[0040] The specific implementation methods of this application will be further described in detail below with reference to the accompanying drawings and embodiments.

[0041] Example 1:

[0042] To investigate the traveling wave resonance characteristics of high-speed thin-spread spur gears in aero-engines, this embodiment conducted a traveling wave resonance characteristic experiment on the large gear (driving spur gear) of the afterburner pump gear shaft in an aero-engine accessory casing under ground test conditions. A spur gear pair was selected as the experimental object, and the accuracy of gear resonance prediction was verified. The test system and experimental apparatus included the main body of the afterburner casing tester, a loading system, a lubrication system, a cooling system, a control system, an experimental afterburner casing, a stress measurement system, a vibration measurement system, a derived noise measurement system, a high-speed brush ring system, a speed measurement system, an air compressor, and a data acquisition and analysis system. The main test equipment included... Figure 1 The examples shown are all existing technologies. The explanations are for illustrating the overall methodology. The software used is Abaqus.

[0043] To predict the resonance state of gear traveling waves and address vibration damping issues, this invention accurately models the gear pair, calculates the frequency of the gear pair under interaction, and obtains the final resonance frequency through dynamic modeling and experimental verification. Finally, the resonance speed of the driving spur gear is determined, thereby achieving vibration damping. The specific approach is as follows: Figure 1 As shown. This invention provides a method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory, comprising the following steps:

[0044] Step 1: Establish a three-dimensional model of the gear and perform modal analysis; specifically, the natural frequency of the gear pair under interaction is obtained through static-modal calculation, and then the natural frequency value of the gear is approximated to the dynamic frequency value to predict the resonant speed of the front and rear traveling waves of the driving spur gear and use it as the boundary condition for transient dynamics calculation to calculate the front and rear traveling wave frequencies of the gear.

[0045] Step 1.1: Based on the actual gear geometry parameters and the basic dimensions of the gear shaft, a 3D model of the gear is created using simulation software. The created 3D gear model is then imported into the pre-processing mesh software HyperMesh. C3D8R hexahedral mesh elements are used, and a 3D mesh is generated by mapping 2D meshes. The mesh generation process is as follows: Figure 2 As shown. Mesh independence tests were performed on the modal analysis of the active spur gear, as shown. Figure 3 As shown, when the mesh size is less than 0.7 mm, the modal frequency values ​​of the active spur gear calculated by the finite element method tend to be stable, so this size is selected as the final mesh size.

[0046] Step 1.2: In this embodiment, the gear material properties are taken at 20℃. Considering the interaction of the spur gear pair, static-modal calculations are performed to obtain the gear's natural frequencies; the calculation results are as follows. Figure 4 The following figures show the modal displacement distribution of the first three pitches of the driving spur gear at different frequencies, with natural frequencies of 1118.9Hz, 1817.9Hz, and 4620.5Hz, as shown in (a), (b), and (c), respectively.

[0047] The specific natural frequency of the gear is:

[0048] In a stationary coordinate system, the expressions for the resonant frequencies of the traveling wave before and after resonance in a gear pitch diameter type are:

[0049] ;

[0050] In the formula The gear meshing frequency, The gear rotation frequency, Including the resonant frequency f of the gear traveling wave F and the resonant frequency f of the traveling wave behind the gear. B m is the gear modal pitch number.

[0051] Step 1.3: Based on the traveling wave resonance theory, the gear natural frequency value calculated by modal analysis is approximated to the dynamic frequency value. The forward and backward traveling wave resonance speeds of the driving spur gear are predicted and used as the boundary conditions for transient dynamics calculation. The forward and backward traveling wave resonance frequencies of the gear are then calculated.

[0052] The resonant rotation speed n F,B Including the traveling wave resonant speed Resonance speed with the back-traveling wave Specifically: ;

[0053] The specific resonant frequency is as follows: In a rotating coordinate system, when a gear experiences pitch diameter-type traveling wave resonance, the gear displacement exhibits harmonic characteristics, and the excitation force frequency is equal to the resonant frequencies of the preceding and following traveling waves: ;

[0054] In the formula, k=0.5,1 is the harmonic number of the excitation force, where the amplitude of the excitation force is the largest when k=1, and the natural frequency f of the gear and the frequency f of the excitation force are also related. e When they are equal, traveling wave resonance will occur;

[0055] Example 2:

[0056] To further verify the effectiveness of this method, the following also includes:

[0057] Step 2: Install and arrange the noise sensors; verify the frequency domain characteristics of the active spur gear in the resonance state through noise test experiments, thereby obtaining the actual gear traveling wave resonance frequency.

[0058] In this embodiment, the mounting end cover of the intermediate gear shaft of the booster pump is removed from the auxiliary casing. The internal acoustic waveguide is inserted into the shaft cavity of the intermediate gear shaft. Six holes are evenly machined circumferentially at the mounting positions of the intermediate gear shaft and the large gear, allowing sound waves to be transmitted through the shaft holes to the internal acoustic waveguide, and then through the external acoustic waveguide to the microphone. The location of the acoustic waveguide and the sound propagation path are as follows. Figure 5 As shown, the connection schematic of the noise testing system is as follows: Figure 6 As shown.

[0059] The relevant explanations from the traveling wave resonance theory are as follows:

[0060] When a gear is subjected to an excitation force, it produces two traveling wave vibrations with the same velocity on the left and right sides. When the gear rotates, the velocities of these two traveling waves become different. The one with the same direction of rotation is the forward traveling wave, and the one with the opposite direction is the backward traveling wave.

[0061] Example 3:

[0062] The vibration characteristics of the active spur gear under resonant state are predicted using this method as follows:

[0063] The axial vibration displacement contour plot of the transient dynamics calculation of the active spur gear is basically the same as that of the two- and three-pitch diameter mode shapes, such as... Figure 7 (a)(b)(c) and Figure 8 As shown in (a), (b), and (c), for two-section radial wave resonance, the entire gear can be roughly divided into four sectors. The axial vibration displacement directions of adjacent sectors are opposite, and the vibration displacement exhibits a two-peak, two-valley distribution pattern with a phase difference of 180°. For three-section radial wave resonance, the entire gear can be roughly divided into six sectors. The vibration displacement exhibits a three-peak, three-valley distribution pattern, and the axial vibration displacement directions of adjacent sectors are opposite, with a phase difference of 180°.

[0064] The axial vibration displacement of the driving spur gear under the resonance point condition of a three-section traveling wave was analyzed. The sampling frequency was set to 20 kHz, and the sampling interval was 0.0001 s. Taking the interval from 0.0608 s to 0.0610 s as an example, the axial displacement of the driving spur gear around its circumference was sequentially expanded, and the displacement at each sampling time t1=0.0608s was extracted. , The axial displacement unfolded diagram is as follows Figure 9 As shown, the axial displacement unfolded by the measuring point of the driving spur gear under the three-section traveling wave resonance is a simple harmonic waveform. As the driving spur gear rotates, the position of the reference wave crest changes in the same direction as the gear rotation. The axial vibration displacement unfolded by the three-section traveling wave resonance of the driving spur gear propagates in the form of a wave, that is, the direction of the traveling wave propagation is the same as the direction of gear rotation.

[0065] Time-frequency domain analysis of the measuring point of the driving spur gear (node ​​number 4376015) reveals a periodic fluctuation in the axial displacement of the driving spur gear. Under the two-pitch forward traveling wave resonance state (2320.7 r / min), as... Figure 10 As shown in (a), the axial displacement fluctuates between 0.015 mm and -0.0115 mm. Under the traveling wave resonance state after the two-section diameter (2138.7 r / min), as... Figure 10 As shown in (b), the axial displacement fluctuates between 0.005 mm and -0.005 mm. Frequency domain analysis yields the gear meshing frequency. (1848.00Hz), rotation frequency (39.45Hz) and its second harmonic 2 Traveling wave resonant frequency (1772.5Hz); Back-traveling wave resonant frequency (1824.43Hz), meshing frequency (1749.45Hz) and rotation frequency (37.49Hz) Among them, the relative error in predicting natural frequencies using static-modal theory. ≤5%, as shown in Table 1 below.

[0066] Table 1. Relative errors between theoretical modal predictions and dynamic calculations:

[0067] Figure 11 (a) Time-frequency domain analysis results show that under the predicted three-pitch traveling wave resonance state (6062 r / min), the displacement of the tooth root monitoring point exhibits quasi-steady-state oscillation characteristics, with a significant active spur gear rotation frequency. (100.0Hz), 2 times the frequency (199.9Hz) and meshing frequency (4923.5Hz). In addition to the rotational frequency, a traveling wave vibration frequency also appears. (4623.6Hz) and its frequency division (5223.4Hz).

[0068] based on Figure 11 (b) Time-frequency domain analysis results show that, under the predicted traveling wave vibration speed (5331 r / min) of the three-section drive gear, a significant driving spur gear rotation frequency is observed. (87.5Hz), 2 times the frequency (174.95Hz) and meshing frequency (4323.7Hz); In addition to the rotational frequency, a back-traveling wave vibration frequency also appears. (4561.1Hz) and its frequency division (4036.2Hz);

[0069] Relative error in predicting natural frequencies using static-modal theory ≤2%, as shown in Table 2. This verifies that the traveling wave resonance theory formula is still applicable to aerospace thin-spread spur gears, and also verifies the accuracy of the explicit-dynamic model established in this paper in predicting the resonance frequency.

[0070] Table 2. Relative errors between theoretical modal predictions and dynamic calculations:

[0071] Finally, based on the experiment, the frequency at which the traveling wave of the three-pitch diameter driving spur gear resonates is as follows: Figure 12As shown, the resonant frequency of the traveling wave of the three-section spur gear is 4948.00 Hz, corresponding to a resonant sound pressure amplitude of 26.47 Pa. This, combined with the dynamic prediction of the three-section resonant frequency (4623.6 Hz, with an error of 6.6%), further verifies the reliability of the proposed method.

[0072] Example 4:

[0073] This embodiment proposes an electronic device, including: one or more processors, and a memory, wherein the memory is used to store instructions, and when the instructions are executed by the one or more processors, the one or more processors execute the aforementioned aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory.

[0074] The electronic device may be a mobile phone, computer, or tablet computer, etc., and includes a memory and a processor. The memory stores a computer program, which, when executed by the processor, implements the aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory as described in the embodiments. It is understood that the electronic device may also include input / output (I / O) interfaces and communication components.

[0075] The processor is used to execute all or part of the steps in the aircraft gear resonance prediction and vibration damping method based on traveling wave resonance theory as described in the above embodiments. The memory is used to store various types of data, which may include, for example, instructions for any application or method in the electronic device, as well as application-related data.

[0076] The processor can be implemented as an Application Specific Integrated Circuit (ASIC), Digital Signal Processor (DSP), Programmable Logic Device (PLD), Field Programmable Gate Array (FPGA), controller, microcontroller, microprocessor, or other electronic components, and is used to execute the aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory described in the above embodiments.

[0077] Example 5:

[0078] This embodiment proposes a computer-readable storage medium that stores executable instructions. When these instructions are executed, if they are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium.

[0079] The computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the aviation gear resonance prediction and vibration avoidance method based on traveling wave resonance theory described in the various embodiments of this application.

[0080] The aforementioned storage media include: flash memory, hard disk, multimedia card, card-type memory (e.g., SD (Secure Digital Memory Card) or DX (Memory Data Register, MDR) memory, random access memory (RAM), static random access memory (SRAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), programmable read-only memory (PROM), magnetic memory, disk, optical disk, server, APP (Application) application store, and other media capable of storing program verification codes. These media store computer programs, and when executed by a processor, they can implement the various steps of the aforementioned aviation gear resonance prediction and vibration damping method based on traveling wave resonance theory.

[0081] Example 6:

[0082] This embodiment proposes a computer program product, including a computer program or instructions, which, when executed by a processor, implements the aviation gear resonance prediction and vibration avoidance method based on traveling wave resonance theory.

[0083] Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or part of the technical solution, can be embodied in the form of a computer program product.

[0084] The various embodiments in this application are described in a progressive manner. The same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on describing the differences from other embodiments.

[0085] The scope of protection of this application is not limited to the embodiments described above. Obviously, those skilled in the art can make various modifications and variations to this disclosure without departing from the scope and spirit of this disclosure. If such modifications and variations fall within the scope of the methods disclosed herein and their equivalents, then the intent of this disclosure also includes such modifications and variations.

Claims

1. A method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory, characterized in that, Includes the following steps: Establish a 3D model of the gear and perform modal analysis; Specifically, the natural frequency of the gear pair under interaction is obtained through static-modal calculation. Then, the natural frequency value of the gear is approximated to the dynamic frequency value to predict the resonant speed of the front and rear traveling waves of the driving spur gear and use it as the boundary condition for transient dynamics calculation to calculate the front and rear traveling wave frequencies of the gear.

2. The method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory according to claim 1, characterized in that, Specifically, the following steps are included: Step S1: Based on the actual gear geometry parameters and the basic dimensions of the gear shaft, a three-dimensional gear model is established using simulation software. The established three-dimensional gear model is then imported into the pre-processing mesh software HyperMesh. C3D8R hexahedral mesh elements are used, and a two-dimensional mesh mapping method is employed to generate a three-dimensional mesh for mesh generation. Mesh independence tests are then performed on the modal analysis of the active spur gear. Step S2: Considering the interaction of the spur gear pair, perform static-modal calculations to obtain the natural frequencies of the gears; Step S3: Based on the traveling wave resonance theory, the gear natural frequency value calculated by modal analysis is approximated to the dynamic frequency value. The forward and backward traveling wave resonance speeds of the active spur gear are predicted and used as the boundary conditions for transient dynamics calculation. The forward and backward traveling wave resonance frequencies of the gear are then calculated.

3. The method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory according to claim 2, characterized in that, The specific natural frequency of the gear is: In a stationary coordinate system, the expressions for the resonant frequencies of the traveling wave before and after resonance in a gear pitch diameter type are: ； In the formula The gear meshing frequency, The gear rotation frequency, Including the resonant frequency f of the gear traveling wave F and the resonant frequency f of the traveling wave behind the gear. B m is the number of gear modal pitch diameters.

4. The method for predicting and mitigating resonance in aircraft gears based on traveling wave resonance theory according to claim 3, characterized in that, The resonant rotation speed n F,B Including the traveling wave resonant speed Resonance speed with the back-traveling wave Specifically: ; The specific resonant frequency is as follows: In the rotating coordinate system, the excitation force frequency is equal to the resonant frequencies of the preceding and following traveling waves, and thus: ; In the formula, k=0.5,1 is the harmonic number of the excitation force, where the amplitude of the excitation force is the largest when k=1, and the natural frequency f of the gear and the frequency f of the excitation force are also related. e When they are equal, traveling wave resonance will occur.

5. An electronic device, characterized in that, include: One or more processors, and a memory for storing instructions that, when executed by the one or more processors, cause the one or more processors to perform an implementation of the aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory as described in any one of claims 1-4.

6. A computer-readable storage medium, characterized in that, It stores executable instructions that, when executed, cause the processor to perform the aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory as described in any one of claims 1-4.

7. A computer program product, characterized in that, Includes a computer program or instructions that, when executed by a processor, implement the aircraft gear resonance prediction and vibration avoidance method based on traveling wave resonance theory as described in any one of claims 1-4.

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