Propulsion shafting vibration prediction method considering gear backlash bidirectional coupling effect
By correcting the bidirectional coupling relationship between tooth backlash and shaft vibration, a differential equation for propulsion shaft motion is established, solving the problem of insufficient vibration prediction accuracy in existing technologies and achieving high-precision prediction and effective control under nonlinear conditions.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- XIAMEN UNIV
- Filing Date
- 2025-12-19
- Publication Date
- 2026-05-08
AI Technical Summary
Existing technologies have failed to systematically establish a dynamic model of the bidirectional coupling relationship between the nonlinear time-varying characteristics of tooth flank clearance and shaft vibration, resulting in insufficient accuracy in propulsion shaft vibration prediction and limited control effect.
By constructing a propulsion shaft vibration prediction method based on the bidirectional coupling effect of tooth flank clearance, the bidirectional coupling relationship between tooth flank clearance and shaft vibration is corrected. The time-varying meshing stiffness and transmission error are corrected using the tooth flank clearance function. The motion differential equation of the propulsion shaft is established, and the tooth flank clearance-shaft vibration characteristics are solved iteratively.
It improves the accuracy of vibration prediction and the realism of response prediction under nonlinear conditions, and can accurately assess the nonlinear dynamic behavior under variable load and variable speed conditions, thereby enhancing the vibration suppression effect.
Smart Images
Figure CN121997449A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the technical field of propulsion shaft systems, and in particular to a method for predicting propulsion shaft system vibration considering the bidirectional coupling effect of tooth flank clearance. Background Technology
[0002] The propulsion shafting is a core component of a ship's power system, and its vibration characteristics directly affect the ship's overall reliability, passenger comfort, and stealth capabilities. This vibration primarily originates from the excitation forces generated by the operation of power equipment, such as diesel engines or gas turbines. These excitation forces are transmitted to the hull through the shafting, bearings, and support structures, ultimately causing engine vibration and radiating noise outwards.
[0003] In complex vibration transmission paths, the gearbox, as a key transmission node in a power transmission device, has a decisive influence on system vibration due to its dynamic characteristics. In particular, nonlinear factors such as tooth backlash and time-varying meshing stiffness can induce strong dynamic excitation, generating shaft frequency and meshing frequency vibration components related to the number of teeth and rotational speed. More importantly, there is a two-way coupling effect between tooth backlash and shaft vibration: on the one hand, the size of the tooth backlash changes the actual meshing position of the gears, thus affecting the time-varying meshing stiffness, and consequently affecting the transverse-longitudinal-torsional coupled vibration of the shaft system; on the other hand, the transverse-longitudinal-torsional coupled vibration of the shaft system can lead to the relative displacement of the gear pair, thereby changing the effective tooth backlash variation and exacerbating the nonlinear characteristics of the tooth backlash.
[0004] This two-way coupling effect of gap and vibration is particularly prominent under variable speed and load conditions, which may lead to nonlinear dynamic behaviors such as bifurcation and chaos in the system, significantly aggravating vibration and noise levels.
[0005] Currently, shaft vibration control technology mainly employs two categories: active and passive control techniques. For example, ultra-low frequency vibration control is achieved by adjusting air chamber pressure, or vibration transmission is suppressed by installing rubber vibration isolators or floating raft isolation systems between the power equipment and the machine body. However, existing methods often focus on suppressing linear vibration components or locally intervening in the transmission path, or only compensating for linear vibration components. They lack a systematic dynamic model that describes the bidirectional coupling relationship between the nonlinear time-varying characteristics of tooth flank clearance and shaft vibration, resulting in insufficient accuracy in propulsion shaft vibration prediction and limited control effectiveness. This leads to this case. Summary of the Invention
[0006] The purpose of this invention is to provide a propulsion shaft vibration prediction method that considers the bidirectional coupling effect of tooth flank clearance. The technical problem to be solved is to provide a propulsion shaft vibration prediction method that incorporates tooth flank clearance into the propulsion shaft vibration calculation and uses the vibration calculation results at the current moment to correct the tooth flank clearance at the next moment, thereby realizing the bidirectional coupling of tooth flank clearance and shaft vibration.
[0007] To achieve the above objectives, the solution of the present invention is: a method for predicting the vibration of a propulsion shaft system considering the bidirectional coupling effect of tooth flank clearance; comprising the following steps: S1, based on the initial tooth backlash at time t, the time-varying meshing stiffness and transmission error of the helical gear at that time are corrected by the tooth backlash function to obtain the first time-varying meshing stiffness and effective transmission error corrected by the tooth backlash; S2, using the first time-varying meshing stiffness and effective transmission error to correct the relevant parameters of the gear meshing unit, so as to establish the differential equation of motion of the propulsion shaft system after tooth backlash correction, solve the differential equation of motion of the propulsion shaft system, and obtain the vibration displacement of each node of the propulsion shaft system at time t. S3, the tooth flank clearance is corrected by the vibration displacement of the propulsion shaft system at time t to obtain the initial tooth flank clearance at time t+1. Steps S1 to S3 are repeated to obtain the vibration characteristics of the propulsion shaft system at any time with bidirectional coupling between tooth flank clearance and shaft vibration.
[0008] Furthermore, in step S2, the relevant parameters of the gear meshing unit include the stiffness matrix, damping matrix, and excitation column vector of the gear meshing unit. The stiffness matrix and damping matrix of the gear meshing unit are corrected using the first time-varying meshing stiffness. The excitation column vector of the gear meshing unit is corrected using the first time-varying meshing stiffness and the effective transmission error; The stiffness matrix of the modified gear meshing element Damping matrix of gear meshing unit and gear meshing unit excitation column vector for: ; ; ; In the formula, V is the column vector of displacement projections generated by the displacement components of each degree of freedom of the gear meshing unit along the meshing line of the helical gear, and ζ m The meshing damping ratio, For the mass of the driving wheel, The mass of the driven gear, Let be the first time-varying meshing stiffness at time t. Let be the effective propagation error at time t.
[0009] Furthermore, in step S1, the calculation method for the first time-varying meshing stiffness includes the following steps: S101, Calculate the time-varying meshing stiffness of the helical gear at time t, neglecting the effect of tooth flank backlash. ; S102, Based on the meshing force F under a given torque, calculate the transmission error of each thin-plate gear pair at time t, without considering the influence of tooth backlash; ; in, This represents the meshing force on the i-th sprocket pair of the j-th simultaneously meshing gear pair at time t. It is obtained by dividing the meshing force F under a given torque by the number of sprocket pairs in contact at time t. Let be the transmission error of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t. Let be the time-varying meshing stiffness of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t; The maximum value among the transmission errors of each thin-plate gear pair is taken as the transmission error of the helical gear. ; S103 introduces a backlash function to address the transmission error of helical gears. After correction, the first transmission error is obtained. ; S104, based on step S103, re-determine the actual contact condition of each thin-plate gear pair, when When the stiffness contribution coefficient of the corresponding gear pair is 0, the stiffness contribution coefficient is 1 in all other cases. S105, Calculate the actual meshing force of the helical gear. , ; in, Let be the time-varying meshing stiffness of the i-th thin-plate gear pair corresponding to the j-th simultaneously meshing gear pair at time t. Let be the stiffness contribution coefficient corresponding to the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t. The first transmission error of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t after tooth backlash correction; Actual meshing force The first transmission error is determined by comparing the meshing force F under a given torque with the relative error between the two, and if the relative error between the two is less than or equal to the allowable error threshold. That is, the effective transmission error When the relative error between the two exceeds the allowable error threshold, the first error is re-propagated. Make corrections until the corrected first transmission error is reached. Able to make The relative error between F and F is less than or equal to the allowable error threshold. In this case, the corrected first transmission error... To effectively transmit errors , in order to pass and effective transmission error To calculate the first time-varying meshing stiffness; ; in, To effectively transmit errors, F represents the actual meshing force.
[0010] Furthermore, in step S105, the first transmission error The correction method is as follows: ; The corrected first propagation error Substitute into step S105 and recalculate the actual meshing force. until The relative error between F and F is less than or equal to the allowable error threshold.
[0011] Furthermore, in step S101, the time-varying meshing stiffness of the helical gear at time t is calculated: ; Where Ceil(ε) represents the smallest integer not less than the total overlap ratio ε, and n is the total number of pairs of thin-plate gears with a single pair of teeth. Let be the stiffness contribution coefficient corresponding to the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t, where the stiffness contribution coefficient corresponding to the non-meshing thin-plate gear pair is 0, and the rest are 1; in, Let be the time-varying meshing stiffness of the i-th thin-plate gear pair corresponding to the j-th simultaneously meshing gear pair at time t. Among them, the meshing stiffness of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair The calculation method is as follows: Without considering tooth flank clearance, the helical gear is divided into several sliced micro-elements using the slicing method. Each sliced micro-element can be regarded as a spur gear. The meshing stiffness of each sliced spur gear is calculated using the potential energy method. Then, the meshing stiffness of the i-th sliced gear pair of the j-th simultaneously meshing tooth pair is: ; In the formula, k h For Hertzian contact stiffness, , , , The following are, in order: bending stiffness, shear stiffness, radial compressive stiffness, and matrix deformation stiffness of the driving wheel. , , , The following are, in order: bending stiffness, shear stiffness, radial compressive stiffness, and matrix deformation stiffness of the driven wheel.
[0012] Furthermore, in step S103, a tooth flank clearance function is introduced to address the transmission error in step S102. After correction, the first transmission error is obtained. ; ; Among them, b i,j This represents half of the initial tooth flank clearance of the i-th thin-plate gear pair of the j-th simultaneously meshing tooth pair at that moment.
[0013] Furthermore, in step S3, the calculation method based on the initial tooth flank clearance at time t+1 includes the following steps: S301, extract the vibration displacement at the gear meshing node of the propulsion shaft system at time t, and calculate the correction term for the gear backlash of different thin-plate gear pairs under translational degrees of freedom. ; S302, Calculate the correction term for the backlash of different thin-plate gear pairs at time t under rotational degrees of freedom. ; S303, corrects the tooth backlash at time t, which is the initial tooth backlash of each thin-plate gear pair at time t+1; ; in, Let t be the initial tooth flank clearance of the helical gear pair at time t.
[0014] Furthermore, the correction term for the gear backlash of different thin-plate gear pairs at time t under translational degrees of freedom. The calculation method is as follows: Establish a global coordinate system OXYZ for the propulsion shaft system. This global coordinate system OXYZ takes the geometric center of the driving wheel as the origin O, the axial direction of the driving gear as the Z-axis, and the line connecting the centers of the gear pair as the X-axis, where the direction from the driving wheel to the driven wheel is the positive direction of the X-axis. The axis perpendicular to the XOZ plane is taken as the Y-axis. The actual center distance of the gear pair in the XOY plane, considering the effects of tooth backlash and shaft vibration, is obtained. ; ; in, Let be the initial center distance of the gear pair in the XOY plane. , These represent the vibration displacements of the driving and driven wheels in the X direction, respectively. , These represent the vibration displacements of the driving and driven wheels in the Y direction, respectively. Calculate the actual end-face pressure angle at this time: ; in, , These are the base circle radii of the driving and driven gears, respectively. Therefore, the correction term for the tooth flank clearance under the translational degrees of freedom at that moment can be obtained: ; in, The initial end-face pressure angle of the gear pair. .
[0015] Furthermore, the correction term for the tooth flank backlash of different thin-plate gear pairs at time t under rotational degrees of freedom. The calculation method is as follows: ; in, Let Z be the Z-axis coordinate value of the geometric center of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair. , These represent the rotation angles of the driving and driven wheels as they rotate counterclockwise around the Y-axis.
[0016] After adopting the above solution, the beneficial effects of the present invention are as follows: First, by constructing a dynamic correction and iterative solution of "clearance-stiffness / error-vibration-clearance", the present invention couples the nonlinear factors of tooth clearance into the shaft dynamics model, which can take into account the iterative solution efficiency of the model, accurately and quickly predict the vibration characteristics of the propulsion shaft, and improve the prediction accuracy under nonlinear working conditions. Secondly, by updating the tooth flank clearance in real time based on vibration displacement, a two-way coupling simulation is achieved between the tooth flank clearance and the vibration of the shaft system's translational and rotational degrees of freedom. This enables accurate evaluation of nonlinear dynamic behaviors of gears that may occur under variable load, variable speed, or impact conditions, such as tooth dislodgement and reverse impact, thus improving the authenticity and reliability of vibration response prediction. In addition, the coupled dynamics model established by this invention truly reflects the state of the propulsion shaft system under nonlinear excitation, providing accurate input loads and boundary conditions for subsequent optimization of the main vibration system parameters and evaluation of gear modification schemes, which helps to improve the vibration suppression and dynamic design of the entire propulsion system. Attached Figure Description
[0017] Figure 1 This is a schematic diagram of the tooth flank clearance of the present invention.
[0018] Figure 2 This is a schematic diagram of the overall matrix construction of the propulsion shaft system of the present invention.
[0019] Figure 3 This is a schematic diagram of the global coordinate system of the propulsion shaft system of the present invention.
[0020] Figure 4 This is a flowchart of the bidirectional coupling correction process for tooth flank clearance and shaft vibration according to the present invention. Detailed Implementation
[0021] This invention provides a method for predicting propulsion shaft vibration considering the bidirectional coupling effect of tooth flank clearance, comprising the following steps: S1, based on the initial tooth backlash at time t, the time-varying meshing stiffness and transmission error of the helical gear at that time are corrected by the tooth backlash function to obtain the first time-varying meshing stiffness and effective transmission error corrected by the tooth backlash; S2, using the first time-varying meshing stiffness and effective transmission error to correct the relevant parameters of the gear meshing unit, so as to establish the differential equation of motion of the propulsion shaft system after tooth backlash correction, solve the differential equation of motion of the propulsion shaft system, and obtain the vibration displacement of each node of the propulsion shaft system at time t. S3, the tooth flank clearance is corrected by the vibration displacement of the propulsion shaft system at time t to obtain the initial tooth flank clearance at time t+1. Steps S1 to S3 are repeated to obtain the vibration characteristics of the propulsion shaft system at any time with bidirectional coupling between tooth flank clearance and shaft vibration.
[0022] In step S1, the calculation method for the first time-varying meshing stiffness includes the following steps: Step S101: Without considering tooth flank backlash, calculate the time-varying meshing stiffness of the helical gear at time t: ; Where Ceil(ε) represents the smallest integer not less than the total overlap ratio ε, and n is the total number of pairs of thin-plate gears with a single pair of teeth. Let be the stiffness contribution coefficient corresponding to the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t, where the stiffness contribution coefficient corresponding to the non-meshing thin-plate gear pair is 0, and the rest are 1; in, Let be the time-varying meshing stiffness of the i-th thin-plate gear pair corresponding to the j-th simultaneously meshing gear pair at time t; Specifically, without considering tooth flank clearance, the helical gear is divided into several sliced micro-elements using the slicing method. Each sliced micro-element can be regarded as a spur gear. The meshing stiffness of each sliced spur gear is calculated using the potential energy method. Then, the meshing stiffness of the i-th sliced gear pair of the j-th simultaneously meshing tooth pair is: : ; Where, k h For Hertzian contact stiffness, , , , The following are, in order: bending stiffness, shear stiffness, radial compressive stiffness, and matrix deformation stiffness of the driving wheel. , , , The following are, in order: bending stiffness, shear stiffness, radial compressive stiffness, and matrix deformation stiffness of the driven wheel.
[0023] Step S102: Without considering tooth flank backlash, calculate the transmission error of the corresponding thin-plate gear pair at time t: ; in, Let represent the meshing force on the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t. The meshing force F under a given torque is obtained by dividing the number of pairs of thin-plate gears in contact at time t. Let be the time-varying meshing stiffness of the i-th thin-plate gear pair corresponding to the j-th simultaneously meshing gear pair at time t. Let be the transmission error of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t; Furthermore, the maximum value among the transmission errors of each thin-plate gear pair is taken; this maximum value is the first transmission error of the helical gear. ; Step S103, focusing on Figure 1 As shown, a tooth flank clearance function is introduced to address the first transmission error in step S102. The correction is performed to obtain the first transmission error after tooth flank clearance correction. ; ; Among them, b i,j This represents half of the initial tooth flank clearance of the i-th thin-plate gear pair of the j-th simultaneously meshing tooth pair at that moment; It represents the first transmission error of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at that moment.
[0024] Step S104: Based on step S103, the relationship between transmission error and tooth backlash is reassessed for each thin-plate gear pair individually to obtain the actual contact condition of each thin-plate gear pair after tooth backlash correction. The stiffness contribution coefficient vector of each thin-plate gear pair is then corrected based on the actual contact condition. At this moment, when... When the stiffness contribution coefficient is 0, it indicates that the thin-plate gear pair is not engaged; otherwise, the stiffness contribution coefficient is 1. Step S105, based on the first transmission error in step S103 The stiffness contribution coefficient in step S104 and the time-varying meshing stiffness in step S101 are used to calculate the actual meshing force of the helical gear at that moment. ; ; The actual meshing force Compare with the meshing force F at a given torque; When the relative error between the two is less than or equal to the allowable error threshold, then the first transmission error... That is, the effective transmission error , in order to pass and effective transmission error To calculate the first time-varying meshing stiffness; When the relative error between the two exceeds the allowable error threshold, the first error is re-propagated. Make corrections until the corrected first transmission error is reached. Able to make The relative error between F and F is less than or equal to the allowable error threshold. In this case, the corrected first transmission error... For effective transmission error , in order to pass and effective transmission error To calculate the first time-varying meshing stiffness; ; in, For effective transmission error, F is the actual meshing force.
[0025] The effective transmission error is such that it enables... The relative error between F and F is less than or equal to the allowable threshold propagation error; Specifically, the relative error is expressed as ,when At 1%, for the first propagation error Make corrections; ; in, This represents the time-varying meshing stiffness of the helical gear at that moment without backlash correction.
[0026] when At 1%, first address the first propagation error. Make corrections, and pass the corrected first transmission error. To calculate the actual meshing force The corrected actual meshing force is obtained. Then the actual meshing force Compare with the meshing force F at a given torque until the corrected actual meshing force is obtained. The relative error between F and F 1%.
[0027] In step S2, establishing the differential equation of motion for the propulsion shaft system corrected for tooth backlash includes the following steps: S201, firstly, the differential equations of motion of the propulsion shaft system without tooth backlash correction are established using the finite node method: ; in, Here is the system inertia matrix. Here is the system stiffness matrix. Here is the system damping matrix. Let be the system excitation column vector. , , The columns of vibration acceleration, vibration velocity, and vibration displacement of each node in the overall propulsion shaft system are, in order. The finite node method is used to perform structural analysis on the propulsion shafting of a ship. The propulsion shafting is divided into shaft segment elements and gear meshing elements. The mass matrix, stiffness matrix and damping matrix of each element are established, and the kinematic differential equation of the propulsion shafting is established through the above matrices.
[0028] S202, the relevant parameters of the gear meshing unit include the stiffness matrix, damping matrix, and excitation column vector of the gear meshing unit. The stiffness matrix and damping matrix of the gear meshing unit are corrected using the first time-varying meshing stiffness; the excitation column vector of the gear meshing unit is corrected using the first time-varying meshing stiffness and the effective transmission error. The corrected stiffness matrix of the gear meshing unit is... Damping matrix of gear meshing unit and gear meshing unit excitation column vector As shown below: ; ; ; In the formula, V is the column vector of displacement projections generated by the displacement components of each degree of freedom of the gear meshing unit along the meshing line of the helical gear, and ζ m The meshing damping ratio, For the mass of the driving wheel, The mass of the driven gear, Let be the first time-varying meshing stiffness at time t. The effective propagation error at time t; S203, focusing on Figure 2 As shown, the stiffness matrix [K'] of each of the above-mentioned modified gear meshing units is... m (t)], damping matrix of gear meshing unit [C' m [t] is decomposed into four sub-matrices of the same size, and the gear meshing unit excitation column vector {F' m(t)} is decomposed into two sub-column vectors of the same size and synthesized into the stiffness matrix, damping matrix and excitation column vector of the overall propulsion shaft system to obtain the propulsion shaft system motion differential equations corrected for tooth backlash; ; in, , These are the overall propulsion shaft system stiffness matrix and damping matrix after considering tooth flank clearance correction. The overall propulsion shaft system excitation column vector is considered after tooth flank clearance correction.
[0029] S204. The Newmark time-domain method is used to solve the differential equation of motion of the propulsion shaft system after tooth backlash correction, and the vibration characteristics of each node of the propulsion shaft system at time t are obtained, including vibration displacement, velocity and acceleration, etc. The calculation by the Newmark time-domain method is well known and will not be described in detail.
[0030] Step S3: Correct the tooth flank clearance by the vibration displacement of the propulsion shaft system at time t to obtain the initial tooth flank clearance at time t+1. Repeat steps S1 to S3 to obtain the vibration characteristics of the propulsion shaft system at any time after tooth flank clearance correction, where time t+1 is the next time after time t.
[0031] Key points combined Figure 3 As shown, the vibration displacement at the gear meshing node of the propulsion shaft system at time t is extracted, and a global coordinate system OXYZ for the propulsion shaft system is established. This global coordinate system OXYZ takes the geometric center of the driving gear as the origin O, the axial direction of the driving gear as the Z-axis, and the line connecting the centers of the gear pair as the X-axis, where the direction from the driving gear to the driven gear is the positive direction of the X-axis. The axis perpendicular to the XOZ plane is taken as the Y-axis. The actual center distance of the gear pair in the XOY plane considering the effects of tooth backlash and shaft vibration is obtained. : ; in, Let be the initial center distance of the gear pair in the XOY plane. This represents the vibration displacement of the driving wheel in the X direction. The vibration displacement of the driven wheel in the X direction is... This represents the vibration displacement of the driving wheel in the Y direction. The driven wheel's vibration displacement in the Y direction; Calculate the actual end-face pressure angle at this time: ; in, Let the base circle radius of the driving wheel be . Let be the base circle radius of the driven gear; Then, at that moment, the correction term for the tooth flank clearance under the translational degrees of freedom is: ; in, The initial end-face pressure angle of the gear pair. .
[0032] Furthermore, when considering the rotational degree of freedom about the Y-axis, the correction term for the tooth flank backlash of different thin-plate gear pairs at time t under the rotational degree of freedom is calculated: ; in, Let Z be the Z-axis coordinate value of the geometric center of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair. , These represent the rotation angles of the driving and driven wheels as they rotate counterclockwise around the Y-axis.
[0033] The tooth flank clearance at time t is corrected to obtain the initial tooth flank clearance of each thin-plate gear pair at the next time step: ; in, Let be the initial tooth flank clearance of the helical gear pair at time t. This represents the correction term for the gear backlash of different thin-plate gear pairs at time t under translational degrees of freedom. The correction term for tooth backlash of different thin-plate gear pairs at time t under rotational degrees of freedom.
[0034] Key points combined Figure 4 As shown, the initial tooth backlash of the helical gear pair at time t+1 is substituted into step S1 to calculate the vibration characteristics of the propulsion shaft system at the next time t+1. By repeating steps S1 to S3, the vibration characteristics of the propulsion shaft system considering the two-way coupling of tooth backlash and shaft vibration at any time can be obtained, and the vibration prediction of the propulsion shaft system considering the two-way coupling effect of tooth backlash can be realized.
Claims
1. A method for predicting propulsion shaft vibration considering the bidirectional coupling effect of tooth flank clearance, characterized in that: Includes the following steps: S1, based on the initial tooth backlash at time t, the time-varying meshing stiffness and transmission error of the helical gear at that time are corrected by the tooth backlash function to obtain the first time-varying meshing stiffness and effective transmission error corrected by the tooth backlash; S2, using the first time-varying meshing stiffness and effective transmission error to correct the relevant parameters of the gear meshing unit, so as to establish the differential equation of motion of the propulsion shaft system after tooth backlash correction, solve the differential equation of motion of the propulsion shaft system, and obtain the vibration displacement of each node of the propulsion shaft system at time t. S3, the tooth flank clearance is corrected by the vibration displacement of the propulsion shaft system at time t to obtain the initial tooth flank clearance at time t+1. Steps S1 to S3 are repeated to obtain the vibration characteristics of the propulsion shaft system at any time with bidirectional coupling between tooth flank clearance and shaft vibration.
2. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 1, characterized in that: In step S2, the relevant parameters of the gear meshing unit include the stiffness matrix, damping matrix, and excitation column vector of the gear meshing unit; The stiffness matrix and damping matrix of the gear meshing unit are corrected using the first time-varying meshing stiffness. The excitation column vector of the gear meshing unit is corrected using the first time-varying meshing stiffness and the effective transmission error; The stiffness matrix of the modified gear meshing element Damping matrix of gear meshing unit and gear meshing unit excitation column vector for: ； ； ； In the formula, V is the column vector of displacement projections generated by the displacement components of each degree of freedom of the gear meshing unit along the meshing line of the helical gear, and ζ m The meshing damping ratio, For the mass of the driving wheel, The mass of the driven gear, Let be the first time-varying meshing stiffness at time t. Let be the effective propagation error at time t.
3. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 1, characterized in that: In step S1, the calculation method for the first time-varying meshing stiffness includes the following steps: S101, Calculate the time-varying meshing stiffness of the helical gear at time t, neglecting the effect of tooth flank backlash. ; S102, Based on the meshing force F under a given torque, calculate the transmission error of each thin-plate gear pair at time t, without considering the influence of tooth backlash; ； in, This represents the meshing force on the i-th sprocket pair of the j-th simultaneously meshing gear pair at time t. It is obtained by dividing the meshing force F under a given torque by the number of sprocket pairs in contact at time t. Let be the transmission error of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t. Let be the time-varying meshing stiffness of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t; The maximum value among the transmission errors of each thin-plate gear pair is taken as the transmission error of the helical gear. ; S103 introduces a backlash function to address the transmission error of helical gears. After correction, the first transmission error is obtained. ; S104, based on step S103, re-determine the actual contact condition of each thin-plate gear pair, when When the stiffness contribution coefficient of the corresponding gear pair is 0, the stiffness contribution coefficient is 1 in all other cases. S105, Calculate the actual meshing force of the helical gear. , ； in, Let be the time-varying meshing stiffness of the i-th thin-plate gear pair corresponding to the j-th simultaneously meshing gear pair at time t. Let be the stiffness contribution coefficient corresponding to the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t. The first transmission error of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t after tooth backlash correction; Actual meshing force The first transmission error is determined by comparing the meshing force F under a given torque with the relative error between the two, and if the relative error between the two is less than or equal to the allowable error threshold. That is, the effective transmission error When the relative error between the two exceeds the allowable error threshold, the first error is re-propagated. Make corrections until the corrected first transmission error is reached. Able to make The relative error between F and F is less than or equal to the allowable error threshold. In this case, the corrected first transmission error... To effectively transmit errors , in order to pass and effective transmission error To calculate the first time-varying meshing stiffness; ； in, To effectively transmit errors, F represents the actual meshing force.
4. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 3, characterized in that: In step S105, the first transmission error The correction method is as follows: ； The corrected first propagation error Substitute into step S105 and recalculate the actual meshing force. until The relative error between F and F is less than or equal to the allowable error threshold.
5. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 3, characterized in that: In step S101, the time-varying meshing stiffness of the helical gear at time t is calculated: ； Where Ceil(ε) represents the smallest integer not less than the total overlap ratio ε, and n is the total number of pairs of thin-plate gears with a single pair of teeth. Let be the stiffness contribution coefficient corresponding to the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair at time t, where the stiffness contribution coefficient corresponding to the non-meshing thin-plate gear pair is 0, and the rest are 1; in, Let be the time-varying meshing stiffness of the i-th thin-plate gear pair corresponding to the j-th simultaneously meshing gear pair at time t. Among them, the meshing stiffness of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair The calculation method is as follows: Without considering tooth flank clearance, the helical gear is divided into several sliced micro-elements using the slicing method. Each sliced micro-element can be regarded as a spur gear. The meshing stiffness of each sliced spur gear is calculated using the potential energy method. Then, the meshing stiffness of the i-th sliced gear pair of the j-th simultaneously meshing tooth pair is: ； In the formula, k h For Hertzian contact stiffness, , , , The following are, in order: bending stiffness, shear stiffness, radial compressive stiffness, and matrix deformation stiffness of the driving wheel. , , , The following are, in order: bending stiffness, shear stiffness, radial compressive stiffness, and matrix deformation stiffness of the driven wheel.
6. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 3, characterized in that: In step S103, a tooth flank clearance function is introduced to address the transmission error in step S102. After correction, the first transmission error is obtained. ; ； Among them, b i,j This represents half of the initial tooth flank clearance of the i-th thin-plate gear pair of the j-th simultaneously meshing tooth pair at that moment.
7. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 1, characterized in that: Step S3, based on the calculation method for the initial tooth flank clearance at time t+1, includes the following steps: S301, extract the vibration displacement at the gear meshing node of the propulsion shaft system at time t, and calculate the correction term for the gear backlash of different thin-plate gear pairs under translational degrees of freedom. ; S302, Calculate the correction term for the backlash of different thin-plate gear pairs at time t under rotational degrees of freedom. ; S303, corrects the tooth backlash at time t, which is the initial tooth backlash of each thin-plate gear pair at time t+1; ； in, Let t be the initial tooth flank clearance of the helical gear pair at time t.
8. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 7, characterized in that: The correction term for the gear backlash of different thin-plate gear pairs at time t under translational degrees of freedom. The calculation method is as follows: A global coordinate system OXYZ is established for the propulsion shaft system. This system has its origin O at the geometric center of the driving gear, its Z-axis at the axis of the driving gear, and its X-axis at the line connecting the centers of the gear pairs. The positive direction of the X-axis is the direction from the driving gear to the driven gear. The Y-axis is the axis perpendicular to the XOZ plane. This yields the actual center distance of the gear pairs in the XOY plane, considering the effects of tooth backlash and shaft vibration. ; ； in, Let be the initial center distance of the gear pair in the XOY plane. , These represent the vibration displacements of the driving and driven wheels in the X direction, respectively. , These represent the vibration displacements of the driving and driven wheels in the Y direction, respectively. Calculate the actual end-face pressure angle at this time: ； in, , These are the base circle radii of the driving and driven gears, respectively. Therefore, the correction term for the tooth flank clearance under the translational degrees of freedom at that moment can be obtained: ； in, The initial end-face pressure angle of the gear pair. .
9. The propulsion shaft vibration prediction method considering the bidirectional coupling effect of tooth flank clearance as described in claim 7, characterized in that: The correction term for the tooth backlash of different thin-plate gear pairs at time t under rotational degrees of freedom The calculation method is as follows: ; in, Let Z be the Z-axis coordinate value of the geometric center of the i-th thin-plate gear pair of the j-th simultaneously meshing gear pair. , These represent the rotation angles of the driving and driven wheels as they rotate counterclockwise around the Y-axis.