A bearing misalignment underformed gear rotor system vibration prediction method

Abstract

The application provides a bearing misalignment and underformed gear rotor system vibration prediction method, and the technical scheme is as follows: a slice deviation is calculated according to a misalignment angle and a modification parameter; initial stiffness is determined based on equivalent cantilever, base and contact flexibility series connection; a slice shear coupling coordination load distribution is introduced; a partial load torsional stiffness is calculated and is combined with parallel slice stiffness series connection to form time-varying meshing stiffness; a nonlinear bearing model is established to determine supporting stiffness; system equations are assembled and time domain iteration solving is carried out, and in a time step, system states are updated in real time according to instantaneous response to obtain dynamic response. The application can solve the technical problem that the full coupling nonlinear vibration response of the system under non-ideal working conditions cannot be accurately and efficiently simulated due to the preset stiffness and supporting state.

Description

Technical Field

[0001] This invention relates to the field of dynamic modeling technology for mechanical transmission systems, specifically to a method for predicting the vibration of a misaligned bearing-modified gear rotor system. Background Technology

[0002] The gear-rotor-bearing system is a core component of rotating machinery transmission systems. Under actual operating conditions, due to factors such as shaft deflection, bearing support displacement, and assembly errors, gear pairs are often in a non-ideal misaligned meshing posture, which can easily lead to uneven loading on the tooth surface, localized stress concentration, and amplified vibration and noise. To mitigate these adverse effects, tooth profile modification techniques are often used in engineering practice to improve the tooth surface contact state.

[0003] However, in existing system dynamics models, time-varying meshing stiffness is usually calculated or set as a pure periodic time function, making it difficult to update the contact state in the tooth width direction in real time based on the instantaneous response of the coupled system in time-domain dynamics simulations. Existing high-precision contact analysis calculations are too costly and difficult to repeatedly call in time-domain simulations; while conventional reduced-order models based on pre-set misalignment conditions cannot accurately characterize the strongly nonlinear evolution of tooth loads during the transition from off-center loading to edge contact. In addition, existing dynamics research has failed to clarify the multidimensional chain coupling mechanism from bearing support variation to system vibration response, severing the dynamic feedback relationship between support variation, shaft deformation, and tooth surface contact state.

[0004] Therefore, how to establish a vibration prediction model that can balance computational efficiency and accurately reflect the real-time coupling feedback of the support and gear meshing state is an urgent problem to be solved in this field. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention proposes a vibration prediction method for a bearing misalignment-modified gear rotor system. This method aims to solve the technical problem that existing technologies cannot accurately and efficiently simulate the fully coupled dynamic evolution and vibration response of the system under non-ideal meshing conditions due to the use of preset parameters for meshing stiffness and bearing support state.

[0006] The technical solution adopted in this invention is as follows: The total deviation of the slice is calculated by obtaining the bearing misalignment angle and tooth profile modification parameters; the initial stiffness of the slice is determined by the series superposition of equivalent cantilever, base, and Hertzian contact compliance; the shear stiffness is calculated by introducing the slice coupling coefficient and iteratively determining the load distribution; the torsional stiffness of the gear tooth topology is determined based on the additional torque and energy method, and synthesized into a comprehensive time-varying meshing stiffness by series synthesis with the equivalent contact stiffness of each slice after parallel connection; a nonlinear bearing model is established to obtain the support stiffness matrix; the time-varying meshing stiffness, support stiffness, and rotor dynamics model are coupled to establish a global equation; a nested algorithm is used for time-domain iterative integration, and the relative misalignment state, meshing stiffness, and support stiffness are updated in real-time and in a closed loop based on the instantaneous response, and the steady-state time-domain response is extracted to predict the vibration characteristics of the system.

[0007] The technical solution of the present invention specifically includes the following steps: Based on the obtained misalignment angle of the bearing and the tooth profile modification parameters of the gear, calculate the total tooth profile deviation of each slice after discrete slicing along the tooth width of the gear pair; Based on the total tooth profile deviation, a dynamic time-varying meshing stiffness model of the gear pair is established, including the continuous shear force coupling between the slices and the three-dimensional topological torsional compliance caused by non-uniform off-center loading. A nonlinear bearing support model is established to obtain the state-dependent dynamic bearing support stiffness that reflects the changes in the contact state of the rolling elements. A rotor dynamics model is established, and the dynamic time-varying meshing stiffness model and bearing dynamic support stiffness are coupled with the rotor dynamics model to construct a global dynamics model of the gear-rotor-bearing system. The global dynamic model is iteratively integrated in the time domain. Within each integration step, the relative misalignment state of the gear pair, the time-varying meshing stiffness, and the dynamic support stiffness of the bearing are updated in real time based on the instantaneous vibration response of the system to obtain the time-domain dynamic response of the system. The vibration characteristics of the system are then predicted based on the time-domain dynamic response.

[0008] Furthermore, based on the obtained bearing misalignment angle and gear tooth profile modification parameters, the total tooth profile deviation of each slice after discrete slicing along the tooth width of the gear pair is calculated. Specific steps include: The gear teeth of the gear pair are uniformly discretized into multiple slices along the tooth width; Based on the misalignment angle of the bearings at the installation positions of the driving and driven gears, calculate the local tooth profile deviation of each slice caused by misalignment. The total tooth profile deviation of each slice is synthesized by combining the local geometric deviations of the slices caused by the tooth profile modification of the gear.

[0009] Furthermore, the tooth profile modification is described using a circular arc curve; The tooth profile modification is a circular arc-shaped modification symmetrical about the midpoint of the tooth width; The local shaping amount of each slice is determined based on the circular arc geometry equation by the radius of the drum-shaped arc and the end shaping amount; The local contact tilt angle of each slice is determined by geometrically coupling and superimposing the relative misalignment angle of the slice with the slope angle of the tangent line at the midpoint of the slice on the circular arc shaping curve.

[0010] Furthermore, the initial integrated contact stiffness of each slice before coupling is determined in the following way: Introducing the equivalent cantilever stress state of each slice, the bending, shearing and axial compression multidimensional compliance components of each slice under meshing axial load are obtained; The deformation compliance of the tooth root matrix and the local Hertzian contact compliance of the tooth surface are coupled, and the multidimensional compliance components are superimposed in series. The initial comprehensive contact stiffness of each slice under local contact deformation is determined by reciprocal transformation.

[0011] Furthermore, in the dynamic time-varying meshing stiffness model, determining the continuous shear force coupling between the slices includes: Establish local interaction relationships between adjacent slices to characterize the shear constraints of the gear tooth entity, and utilize the slice coupling coefficient. Calculate the coupling stiffness between the i-th slice and the (i+1)-th slice. :

[0012] In the formula, Let be the coupling stiffness between the i-th slice and the (i+1)-th slice. The slice coupling coefficient, For the i-th slice, the initial integrated contact stiffness before coupling is not considered. The initial composite contact stiffness of the (i+1)th slice before coupling is considered, where m is the gear module. Let be the slice thickness, and i be the slice number. , where n is the total number of slices; The coupling stiffness is assembled into the slice stiffness matrix of the contact control equation, and deformation coordination iteration including the load redistribution effect between slices is performed.

[0013] Furthermore, in the dynamic time-varying meshing stiffness model, the determination of the three-dimensional topological torsional compliance caused by non-uniform eccentric loading includes: Based on the non-uniform load distribution on each slice, the additional torque generated by the normal force of each slice around the mid-surface of the gear tooth in two orthogonal directions is calculated respectively. The additional torsional strain energy of the gear teeth due to non-uniform eccentric loading in two orthogonal directions is calculated using the deformation energy method, and the topological torsional stiffness of the gear teeth in the two orthogonal directions is determined, which is used to characterize the three-dimensional topological torsional compliance of the gear teeth.

[0014] Furthermore, in the dynamic time-varying meshing stiffness model, the topological torsional stiffness in two orthogonal directions is synthesized along the meshing direction to obtain the total torsional stiffness. ; The equivalent contact stiffness of each slice is calculated and then superimposed in parallel along the tooth width direction. The total equivalent contact stiffness of the superimposed slices is then compared with the total torsional stiffness. By performing series elastic superposition, the overall time-varying meshing stiffness of the gear pair is finally synthesized. The calculation formula is as follows:

[0015] In the formula, To determine the equivalent contact stiffness of the i-th slice after considering the slice coupling effect, The meshing force borne by the i-th slice, Let i be the total deformation of the i-th slice. For the i-th slice deformation component related to the driving pinion, For the i-th slice deformation component related to the driven large gear, Let be the Hertzian contact deformation component of the i-th slice. Let be the combined time-varying meshing stiffness of the gear pair, n be the total number of slices, and i be the slice number. , Let be the total torsional stiffness.

[0016] Furthermore, the nonlinear bearing support model is a quasi-static deep groove ball bearing model; Based on the geometric deformation coordination relationship between the rolling elements and the inner and outer raceways, and considering the three-dimensional force and torque balance relationship of rolling element friction, centrifugal force, and gyroscopic torque, the nonlinear dynamic support force vector acting on the inner ring of the bearing is calculated, and the displacement partial derivative is obtained to obtain the state-related dynamic support stiffness matrix.

[0017] Furthermore, a rotor dynamics model is established, including discretizing the shaft segment using Timoshenko beam elements and establishing the mass matrix, stiffness matrix, and gyro matrix of the shaft segment. The global dynamics model is represented by the following matrix-form control equations:

[0018] In the formula, The global quality matrix of the system. Here is the global damping matrix of the system. This is the system's global gyroscope matrix. Let be the global stiffness matrix of the system, where the global stiffness matrix is... Includes the dynamic support stiffness of the bearing and the combined time-varying meshing stiffness of the gear pair. and rotor shaft stiffness, Let be the generalized displacement vector of the system. Let be the generalized velocity vector of the system, which is The first derivative with respect to time, Let be the generalized acceleration vector of the system, which is The second derivative with respect to time, It is a generalized excitation vector.

[0019] Furthermore, the global dynamics model is subjected to iterative integration in the time domain, and a double-nested time-domain numerical integration is performed using the Newmark-β integration algorithm combined with the Newton-Raphson iterative algorithm. Within each integration time step, the relative misalignment angle of the gear pair under the current step is calculated in real time using the instantaneous translational and rotational responses of the gear nodes. Based on this, the time-varying meshing stiffness and bearing dynamic support stiffness are recalculated and the dynamic state is updated until the dynamic response of the current step meets the convergence condition, and then the next integration time step is entered. The vibration characteristics of the system are predicted based on the time-domain dynamic response, including: The initial transient response in the time-domain dynamic response is removed, and the steady-state vibration response is extracted and subjected to time-domain waveform analysis, frequency-domain spectrum analysis, and response fluctuation statistical analysis to predict the dynamic vibration characteristics of the system under different misalignment and speed conditions.

[0020] As can be seen from the above technical solution, the beneficial technical effects of the present invention are as follows: 1. Accurate simulation of the closed-loop coupling transmission path of the entire system was achieved: It breaks through the limitation of using preset functions for meshing stiffness and support stiffness in traditional models, and realizes the closed-loop dynamic update of the entire physical closed-loop path from bearing support variation to system vibration response in the time domain integration.

[0021] 2. By introducing shear mechanics coupling between adjacent slices and three-dimensional topological torsional stiffness within the slice method framework, the strong nonlinear force characteristics of gear teeth under the transition state from off-center loading to edge contact are accurately characterized while ensuring extremely high dynamic solution efficiency.

[0022] 3. It can accurately capture the dynamic vibration law of the system under different misalignment and speed conditions by using multi-dimensional indicators in the time domain, frequency domain and response fluctuation statistics, providing an efficient and accurate analysis tool for the dynamic design and reliability assessment of the transmission system. Attached Figure Description

[0023] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the accompanying drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. In all the drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn to scale.

[0024] Figure 1 This is a schematic diagram of the discretization of the gear pair meshing plane and tooth slices under misalignment conditions according to an embodiment of the present invention, wherein, Figure 1 (a) is a schematic diagram of the angle displacement of the meshing plane. Figure 1 (b) is a schematic diagram of uniform discretization of the gear tooth slice; Figure 2 This is a schematic diagram of the tooth-direction drum-shaped profile and the discrete tooth slice of an embodiment of the present invention, wherein, Figure 2 (a) is the outline of the circular arc of the modified tooth direction line. Figure 2 (b) is the spatial projection of the reshaped slice; Figure 3 This is a diagram illustrating the force decomposition and geometric cantilever equivalent model of a gear tooth slice according to an embodiment of the present invention. Figure 3 (a) is a three-dimensional decomposition diagram of the meshing force of the slice. Figure 3 (b) is a geometric calculation model diagram of the equivalent cantilever beam under stress; Figure 4 This is a comparison diagram of the equivalent mechanical models of gear tooth stiffness based on slices according to embodiments of the present invention, wherein, Figure 4 (a) is a diagram of the traditional uncoupled slice stiffness model without slice interactions. Figure 4 (b) A diagram of the coupled slice stiffness model considering the coupling effect of slice shear force; Figure 5 This is a schematic diagram of the spatial distribution of forces and moments on the critical section of the gear tooth slice according to an embodiment of the present invention, wherein, Figure 5 (a) is a diagram showing the distribution of elastic torsional moment in the y-direction around the mid-surface of the gear tooth. Figure 5 (b) is a diagram showing the distribution of elastic torsional moment about the x-direction; Figure 6 This is a schematic diagram illustrating the deformation geometric compatibility of the rolling elements in a deep groove ball bearing according to an embodiment of the present invention. Figure 6 (a) is a diagram showing the deformation relationship of a single rolling element within the bearing load-bearing area. Figure 6 (b) is a diagram showing the deformation relationship of a single rolling element in the non-load-bearing region of the bearing; Figure 7 This is a force and torque balance analysis diagram of the inner ring rolling element of the bearing according to an embodiment of the present invention, wherein, Figure 7 (a) is a force balance diagram of a single rolling element in the load-bearing zone. Figure 7(b) is a force diagram of a single rolling element in the non-load-bearing zone; Figure 8 This is a schematic diagram of the overall structure of the spur gear pair-rotor-bearing fully coupled system according to an embodiment of the present invention; Figure 9 This is a schematic diagram of the discretization of shaft segment units and dynamic node division of the bearing rotor system according to an embodiment of the present invention; Figure 10 This is a flowchart illustrating the double-layer nested adaptive iterative solution of the nonlinear time-domain dynamic control equations of the coupled system according to an embodiment of the present invention. Figure 11 This is a comparison diagram in the time and frequency domains of the conventional model, the model of this invention, and the experimental signal in embodiments of the present invention. Figure 11 (a)-(c) are the time-domain acceleration waveforms of the traditional model, the model of this invention, and the experimental signal, respectively. Figure 11 (d)-(f) are the frequency spectra of the traditional model, the model of this invention, and the experimental signal, respectively; Figure 12 This is a three-dimensional waterfall spectrum of the system's variable-speed vibration response under varying bearing misalignment with frequency sweep under variable-speed conditions, assuming tooth width drum-shaped modification according to an embodiment of the present invention. Figure 12 (a) is a three-dimensional waterfall spectrum of the vibration response in the y-direction when the bearing misalignment δ=0mm; Figure 12 (b) is the three-dimensional waterfall spectrum of the vibration response in the z-direction when the bearing misalignment δ=0mm; Figure 12 (c) is the three-dimensional waterfall spectrum of vibration response in the y direction with bearing misalignment δ=0.2mm; Figure 12 (d) is the three-dimensional waterfall spectrum of vibration response in the z-direction when the bearing misalignment δ = 0.2 mm; Figure 12 (e) is a three-dimensional waterfall spectrum of the vibration response in the y-direction when the bearing misalignment δ = 0.4 mm; Figure 12 (f) is a three-dimensional waterfall spectrum of the vibration response in the z-direction with bearing misalignment δ=0.4mm; Figure 12 (g) is a three-dimensional waterfall spectrum of the vibration response in the y direction when the bearing misalignment δ = 0.8 mm; Figure 12 (h) is the three-dimensional waterfall spectrum of vibration response in the z-direction when the bearing misalignment δ = 0.8 mm; Figure 13 This is a box plot showing the statistical distribution of the system's variable-speed vibration response under tooth width drum-shaped modification and different bearing misalignment amounts across the entire speed range, according to an embodiment of the present invention. Figure 13 (a) Box plot of variable speed vibration response in the y direction with bearing misalignment δ=0mm; Figure 13 (b) is a box plot of the variable speed vibration response in the y direction with bearing misalignment δ=0.2mm; Figure 13 (c) is a box plot of the variable speed vibration response in the y direction with bearing misalignment δ=0.4mm; Figure 13 (d) is a box plot of the variable speed vibration response in the y direction with bearing misalignment δ=0.8mm; Figure 13 (e) is a box plot of the variable speed vibration response in the z direction with bearing misalignment δ=0mm; Figure 13 (f) is a box plot of the variable speed vibration response in the z direction with bearing misalignment δ=0.2mm; Figure 13 (g) is a box plot of the variable speed vibration response in the z direction with bearing misalignment δ=0.4mm; Figure 13 (h) is a box plot of the variable speed vibration response in the z direction with bearing misalignment δ=0.8mm. Detailed Implementation

[0025] The embodiments of the technical solution of the present invention will now be described in detail with reference to the accompanying drawings. These embodiments are merely illustrative of the technical solution of the present invention and are therefore intended to limit the scope of protection of the present invention.

[0026] It should be noted that, unless otherwise stated, the technical or scientific terms used in this application should have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0027] This embodiment provides a vibration prediction method for a bearing-misaligned lower modified gear rotor system. The working principle, mathematical derivation of the physical model, and assembly of the system matrix of this embodiment are described in detail below: The method flowchart of this embodiment is as follows: Figure 1 As shown, the specific steps include: Based on the obtained misalignment angle of the bearing and the tooth profile modification parameters of the gear, calculate the total tooth profile deviation of each slice after discrete slicing along the tooth width of the gear pair; Based on the total tooth profile deviation, a dynamic time-varying meshing stiffness model of the gear pair is established, including the continuous shear force coupling between the slices and the three-dimensional topological torsional compliance caused by non-uniform off-center loading. A nonlinear bearing support model is established to obtain the state-dependent dynamic bearing support stiffness that reflects the changes in the contact state of the rolling elements. A rotor dynamics model is established, and the dynamic time-varying meshing stiffness model and bearing dynamic support stiffness are coupled with the rotor dynamics model to construct a global dynamics model of the gear-rotor-bearing system. The global dynamic model is iteratively integrated in the time domain. Within each integration step, the relative misalignment state of the gear pair, the time-varying meshing stiffness, and the dynamic support stiffness of the bearing are updated in real time based on the instantaneous vibration response of the system to obtain the time-domain dynamic response of the system. The vibration characteristics of the system are then predicted based on the time-domain dynamic response.

[0028] In this embodiment, further, based on the obtained misalignment angle of the bearing and the tooth profile modification parameters of the gear, the total tooth profile deviation of each slice after discrete slicing along the tooth width of the gear pair is calculated. The working principle and specific spatial geometric projection process are as follows: In actual operation, due to shaft deflection, bearing support displacement, and assembly errors, the meshing state of gear pairs often deviates from the ideal parallel shaft condition. To describe the relative angular deviation between the primary and driven gears at their installation positions, the relative misalignment angles of the gear pair in two directions are defined as follows:

[0029]

[0030] In the formula, , These are the misalignment angles in the x-direction of the driving and driven gears at their installation positions, respectively. , These represent the misalignment angles in the y-direction for the driving and driven gears at their installation positions. Since the misalignment angle in the y-direction is mainly caused by the nonlinear bearing contact force and its amplitude is relatively small, its influence is ignored in this embodiment; only the misalignment angle in the x-direction of the gear pair is considered. ,like Figure 1 As shown in (a).

[0031] To characterize the local contact variation along the tooth width, such as Figure 1 As shown in (b), the gear teeth are uniformly divided into n slices along the tooth width. The thickness of the i-th slice is... and the coordinates of its center along the tooth width direction Represented as:

[0032]

[0033] In the formula, Let be the thickness of the i-th slice. B represents the number of slices and B represents the tooth width.

[0034] Local tooth profile deviation of the i-th slice caused by misalignment It can be represented as:

[0035] In the formula, For the local tooth profile deviation of the i-th slice, , These represent the angular displacements of the driving and driven wheel nodes in the x-direction, respectively. , These represent the angular displacements of the driving and driven wheel nodes in the y-direction, respectively. It represents the azimuth angle of the meshing line of the two gears in space.

[0036] Under misalignment conditions, the load on the tooth surface tends to concentrate on one side of the tooth width, easily inducing edge contact. To mitigate this effect, this embodiment introduces tooth profile modification. For example... Figure 2 As shown, the tooth profile modification is described by a circular arc curve, and the end profile modification amount is defined as... .

[0037] The radius R of the drum-shaped arc and the local shaping amount of the i-th slice. They are represented as follows:

[0038]

[0039] When considering both misalignment and tooth-direction modification, the local contact angle of the i-th slice can be expressed as:

[0040] In the formula, Let be the relative rotation angle caused by misalignment of the i-th slice. Correspondingly, the total tooth profile deviation of the i-th slice. It is composed of the geometric deviation of the driving wheel slice, the geometric deviation of the driven wheel slice, and the local deviation caused by misalignment:

[0041] In the formula, and These are the local geometric deviations of the i-th slice of the driving gear and the driven gear, respectively, both of which include local tooth profile deviations caused by the modification of the tooth surface arc.

[0042] In this embodiment, the establishment of a dynamic time-varying meshing stiffness model for the gear pair, which includes continuous shear force coupling between structural slices and three-dimensional topological torsional compliance caused by non-uniform off-center loading, is further illustrated by the following steps for mechanical deformation energy derivation and stiffness parallel / serial superposition synthesis: In practical implementation, the initial integrated contact stiffness of a single slice before considering the coupling effect between slices is first analyzed using multidimensional cantilever deformation compliance: like Figure 3 As shown, under misalignment conditions, the local contact postures of each slice along the tooth width are different, causing the direction of the slice meshing force to deviate. Let... The local meshing force of the i-th slice, its tangential component and axial component It can be represented as:

[0043] Tangential component It can be further decomposed into bending components. and axial compression component :

[0044] In the formula, Let be the local contact angle of the i-th slice. The pressure angle at the engagement point.

[0045] Based on such Figure 3 (b) shows the equivalent cantilever beam model, where the bending strain energy of the i-th gear tooth slice is... Shear strain energy and axial compressive strain energy They are represented as follows:

[0046]

[0047]

[0048] In the formula, E and G are the elastic modulus and shear modulus of the material, respectively. and These represent the equivalent cross-sectional areas of the slice at different sections. and These are the corresponding second-axis moments of inertia of the cross sections.

[0049] When considering tooth profile modification, the cross-sectional area and second-order moment of inertia of the i-th slice can be expressed as:

[0050]

[0051] The bending compliance component of the gear tooth slice is derived from the above multidimensional deformation energy equation. Shear compliance component and axial compressive compliance component The integral expression is as follows:

[0052]

[0053]

[0054] Considering the elastic deformation of the tooth root base and the local Hertzian contact deformation of the tooth surface, the corresponding equivalent stiffness can be expressed as follows:

[0055]

[0056] In the formula, For the material's Poisson's ratio, , , , This is an empirical coefficient for matrix deformation. The normal distance from the point of application of the load to the root is . This refers to the width of the tooth root base.

[0057] Based on this, the stiffness of the drive wheel slice can be calculated. From the stiffness of gear slices and the initial integrated contact stiffness of the i-th slice before coupling is not considered. :

[0058]

[0059]

[0060] Based on the above single-slice compliance analysis, we further introduce continuous shear mechanical constraints on the solid structure between slices to construct a coordination relationship of load distribution matrix between adjacent slices that includes local deformation transmission: In fact, the tooth structure is continuous along the tooth width, such as... Figure 4 As shown in (b), there is continuous shear force coupling between adjacent slices. The coupling stiffness between the i-th slice and the (i+1)-th slice is... Represented as:

[0061] In the formula, The slice coupling coefficient is taken as the value in engineering calculations. Where m is the gear module. The continuity constraint of each slice during the deformation transmission process is described by a recursive relation:

[0062]

[0063] in, The deformation transfer coefficient between adjacent slices is given by the following formula:

[0064] When considering both the slice coupling stiffness and the local total tooth profile deviation, the tooth surface contact control equation can be written in matrix form:

[0065] In the above contact control equations, if the gear pair enters the double-tooth meshing region, then two pairs of teeth in the system are simultaneously engaged. In this case, to achieve a coupled iterative solution for the load distribution of the two meshing teeth, the corresponding slice stiffness matrix... Deformation vector Load vector and local error vector It can be further assembled and represented in the form of a block matrix as follows:

[0066] In the formula, the subscripts 1 and 2 of the block correspond to the first pair of meshing teeth and the second pair of meshing teeth in the double-tooth meshing area, respectively, and the block matrix... and These represent the slice stiffness matrices of the first and second meshing tooth pairs, respectively, and the block column vectors. and These represent the deformation vectors of the first and second meshing tooth pairs along the tooth width direction, respectively; block column vectors. and These represent the local contact load vectors assigned to the first and second meshing tooth pairs, respectively, and are block column vectors. and These represent the local geometric error vectors of the first and second meshing tooth pairs caused by reshaping and misalignment, respectively. In each time-domain integration iteration of the double-tooth meshing zone, the total matrix after the above-mentioned block assembly is substituted into the contact compatibility equation for global solution to obtain the dynamic load redistribution law between tooth pairs and between slices.

[0067] To separate the deformation components, the total deformation of the i-th slice is... Decomposed into:

[0068] The corresponding force balance relationship is:

[0069] Furthermore, the contact load borne by all slices should satisfy the following total torque balance condition:

[0070] In the formula, For input torque, The radius of the base circle of the driving wheel.

[0071] Based on the above contact stress and load distribution, we further consider the three-dimensional topological elastic torsional compliance characteristics induced when non-uniform loads generate moments about the gear tooth body in different orthogonal directions: like Figure 5 As shown, due to the combined effects of misalignment and tooth surface modification, the slicing force distributed along the tooth width is uneven, which in turn generates additional torsional moments in the y and x directions:

[0072] Based on the energy method, the elastic torsional deformation energy of the gear teeth in the y and x directions can be expressed as follows:

[0073]

[0074] In the formula, , , , The polar moment of inertia of the corresponding critical section: , Furthermore, the load distribution coefficients of the i-th slice in the x and y directions are introduced. and its corresponding y-axis distribution coefficient :

[0075] Based on this, the topological torsional stiffness k_{ty} and k_{tx} of the gear teeth in two directions are derived:

[0076]

[0077] In the single-tooth meshing region, the total topological torsional stiffness of the meshing pair of gears is determined by the superposition of the torsional compliance of the driving and driven gears in both directions. For the double-tooth meshing region, the two tooth pairs share the load, and their combined total torsional stiffness... It can be represented as: Single-tooth meshing zone: Double-tooth meshing area: In the formula, the subscripts p and g represent the driving gear and the driven gear, respectively, and q represents the q-th meshing tooth pair in the double-tooth meshing area.

[0078] Finally, after the individual slices are stacked in parallel, their equivalent total contact stiffness is then combined with the total torsional stiffness. A series equivalent synthesis is performed to determine the overall time-varying meshing stiffness of the gear pair. The calculation formula is as follows:

[0079] In this embodiment, the specific derivation process of establishing the nonlinear bearing support model to obtain the state-dependent dynamic bearing support stiffness based on the contact pair deformation coordination and torque mechanical balance is as follows: To accurately characterize the strong nonlinear features of the bearing support boundary under variable loads and high-speed dynamic displacements, a quasi-static rolling bearing analysis model is introduced. For example... Figure 6 As shown, when the deep groove ball bearing's first... When a ball is located in the load-bearing area, the geometric deformation compatibility equations between its rolling elements and the inner and outer raceways are expressed as follows:

[0080]

[0081] When the ball enters the non-load-bearing zone, the rolling element loses contact with the raceway, and the deformation compatibility equation degenerates into:

[0082] In the formula, and Let be the position vector component of the ball's center relative to the center of curvature of the inner raceway. and The position components of the outer raceway relative to the center of the sphere. and These are the inner and outer raceway curvature radius coefficients. The diameter of the ball bearing is... , This is due to localized contact deformation.

[0083] like Figure 7 As shown, centrifugal force is introduced under high-speed operating conditions. gyro torque and the coefficient of sliding friction between the rolling elements and the raceway , No. The three-dimensional force and moment balance equations for a rolling element are written as follows:

[0084] In the formula, , The instantaneous normal contact load between the ball and the inner and outer raceways. , This is the local contact angle.

[0085] If the ball is in the non-load-bearing zone, its normal contact force The mechanical equilibrium equations simplify to: .

[0086] After solving the nonlinear equilibrium equations, the nonlinear dynamic support force vector acting on the inner ring of the bearing is synthesized. :

[0087] In the formula, The total number of rolling elements. For the first The angular position of each rolling element The effective lever arm of the contact force. The inner raceway radius is given.

[0088] Force vector relative displacement vector of the inner circle By taking the partial derivative, we can obtain the state-related information. Nonlinear dynamic support stiffness matrix:

[0089] This dynamic support stiffness matrix reflects the influence of the generalized displacement change of the bearing inner ring on the contact load, and can provide state-dependent nonlinear support boundary conditions for the gear-rotor-bearing coupled system. Therefore, the bearing support force and support stiffness are no longer constants, but vary together with the instantaneous displacement of the shaft system, the contact state of the rolling elements, and the rotational speed.

[0090] In this embodiment, further, a rotor dynamics model is established, and the dynamic time-varying meshing stiffness model and the bearing dynamic support stiffness are coupled with the rotor dynamics model to construct a global dynamics model of the gear-rotor-bearing system. This model is then subjected to iterative integration in the time domain. The physical parameters of the pinion and gear, the discretization of the shaft segments, the mass parameters of the spur gear pair Z23 / Z84, and the ball bearing parameters are summarized in Tables 1 to 4, respectively. The configuration of the coupled system and its discretized dynamics model are as follows: Figure 8 and Figure 9 As shown.

[0091] Table 1 Physical parameters of the driving and driven wheels

[0092] Table 2 Division of Shaft Segment Units

[0093] Table 3 Quality parameters of spur gear pair Z23 / Z84

[0094] Table 4 Bearing Parameters

[0095] Each shaft is discretized into 20 shaft elements, totaling 21 nodes. The gear pair is located at node 10, and the four supporting bearings are arranged at nodes 7 and 14 on the driving and driven shafts, respectively. The coupled time-varying meshing stiffness matrix and the nonlinear bearing support stiffness matrix derived above are assembled into the global model. Therefore, the meshing stiffness and bearing support stiffness are no longer constant parameters, but evolve continuously with the instantaneous system response, which makes the coupled system inherently time-varying and nonlinear.

[0096] The specific system global matrix assembly and double-nested adaptive iterative solution process is as follows: The aforementioned dynamic time-varying meshing stiffness model of the gear pair, considering the structural shear force coupling between slices and the three-dimensional topological torsional compliance, the nonlinear bearing support stiffness model, and the rotor finite element model are coupled together to establish the global nonlinear dynamic control equations of the gear-rotor-bearing system:

[0097] In the formula, , , and These are the system's global mass matrix, global damping matrix, global gyroscope matrix, and global stiffness matrix, respectively. Let be the generalized displacement vector of the system. For generalized velocity vectors, It is a generalized acceleration vector. It is a generalized excitation vector.

[0098] The global quality matrix Represented as:

[0099] In the formula, and These are the mass matrices for axis 1 and axis 2, respectively. and These are the mass matrices of the driving pinion and the driven gear, respectively.

[0100] The global stiffness matrix Represented as:

[0101] In the formula, and Here is the stiffness matrix of the shaft segments. This is the meshing stiffness matrix of the spur gear pair, whose diagonal terms are derived from the combined time-varying meshing stiffness. constitute, to These are the nonlinear dynamic support stiffness matrices acting at the four support bearings. , This refers to the stiffness of the coupling.

[0102] The global gyroscope matrix Represented as:

[0103] In the formula, and These are the gyroscope matrices for axis 1 and axis 2, respectively. and These are the gyroscope matrices for the pinion and gear, respectively.

[0104] The system damping matrix is ​​expressed using the Rayleigh proportional damping model as follows:

[0105] In the formula, the mass proportional damping coefficient and stiffness proportional damping coefficient Given the modal damping ratio and the corresponding natural frequency, the following formula is used to determine the damping ratio:

[0106] In the formula, and For the selected first-order and second-order natural frequencies of the system, and This represents the corresponding modal damping ratio.

[0107] The generalized excitation vector experienced by the system Represented as:

[0108] In the formula, and Let represent the excitation vectors introduced by the eccentricity of axis 1 and axis 2, and their own weight, respectively. and These are the excitation vectors of the driving and driven gears caused by mass eccentricity, respectively. This represents the support excitation vector introduced by each bearing due to imbalance and off-center loading.

[0109] The solution process for the system response is as follows: Figure 10 As shown. Within each time step, the system dynamic response is solved using the Newmark-β method. Based on the instantaneous translational displacement and rotational response of the gear nodes, the relative misalignment state of the gear pair and the local tooth profile deviation of each slice are first updated; subsequently, considering the slice coupling effect, the contact control equations are used... The load distribution and deformation compatibility relationship of the slices are solved. Based on this, the local stiffness, tooth torsional stiffness, and overall time-varying meshing stiffness of each slice are recalculated using a slice-based gear model. Therefore, the gear meshing stiffness is not predetermined, but rather iteratively updated in real time at each time step based on instantaneous misalignment, slice deformation, load redistribution, and tooth torsional deformation. Simultaneously, the bearing contact state and support stiffness are also considered. Adaptive updates are performed using a nested Newton-Raphson iterative scheme until the convergence condition is met.

[0110] In this embodiment, the integration parameter is taken as and One rotation of the input axis is defined as one computation cycle, discretized into 5000 integration steps. The total simulation time is 30 revolutions, corresponding to 150,000 integration steps, with a time step size of [missing value]. ,in Let be the instantaneous angular velocity of the input shaft. After removing the initial transient response, the steady-state vibration in the y-direction of the gearbox surface near bearing 4, away from the torque input end of the driven shaft, is selected for subsequent time-domain, frequency-domain, and statistical characteristic analysis.

[0111] In this embodiment, the vibration prediction method for the bearing misalignment and modified gear rotor system proposed in this invention is further tested on an experimental bench through multi-dimensional simulation accuracy benchmarking and model validity verification. The specific process is as follows: A vibration test bench for a gear-rotor-bearing system was established. This test bench mainly consists of a drive motor, gearbox, bearing support structure, and data acquisition system (including accelerometers and data acquisition instruments), providing a reliable experimental basis for comparing the responses of simulation models.

[0112] Figure 11 A comparison of the time-domain waveforms and spectra predicted by the traditional preset stiffness model and the coupled model of this invention with the experimental signals is presented. Figure 11 The traditional model shown in (a) (where the time-varying meshing stiffness uses a preset pure periodic time function) cannot reproduce the complex frequency conversion modulation bands caused by non-ideal misalignment in the measured spectrum, such as... Figure 11 (b) shows obvious spectral line breaks. In contrast, the closed-loop iterative coupling model proposed in this invention shows good agreement with experimental results in terms of the envelope fluctuation form of the time-domain waveform and the distribution of variable spectral bands in the spectrum (e.g., Figure 11 (as shown in (c) and (f)). This demonstrates that the model established in this invention can extremely effectively capture the strongly nonlinear coupled dynamic behavior between gear meshing, nonlinear bearing support, and system dynamic response, exhibiting extremely high simulation accuracy.

[0113] In this embodiment, the sensitivity characteristics of the dynamic response of the gear-rotor-bearing coupling system under different variable speed sweep frequency conditions and different bearing misalignment and off-center load degrees are further discussed and analyzed using parameterization as follows: Rotational speed is a key parameter for controlling the excitation frequency in a gear-rotor-bearing system. As rotational speed increases, the fundamental meshing frequency... Its higher harmonics will continue to move to higher frequencies and eventually excite the structural modes of the system, causing system resonance. This embodiment aims to study the nonlinear coupling relationship between the excitation frequency change caused by rotational speed and the changes in equivalent meshing stiffness and support conditions caused by bearing misalignment. In particular, it aims to determine whether tooth profile modification can play a constant vibration reduction role within the studied rotational speed range, and to study the evolution mechanism of its low-order and high-order natural frequencies when the system is subjected to misalignment. This embodiment uses a constant drum-shaped tooth profile modification amount. Below, a detailed comparison is made of four representative bearing misalignment conditions. , , , Spectral evolution and statistical response characteristics under ( ).

[0114] Figure 12 The three-dimensional waterfall spectrum of the system's variable speed vibration response is shown under different bearing misalignment.

[0115] In fully centered working conditions ( Under these conditions, the vibrational energy of the system is highly concentrated in the main load-bearing direction, i.e., the y-direction (e.g., ...). Figure 12 (a) shows the response at the fundamental engagement frequency. and its harmonic branches ( , The dominant high-frequency resonance band is 1380Hz. When the excitation frequency sweeps past 1380Hz, obvious local resonance peaks are generated, indicating that the region around 1380Hz corresponds to the system's inherent dominant high-frequency resonance band. Because this high-frequency resonance band can be repeatedly excited by multiple meshing harmonics, it exhibits extremely strong modal sensitivity. In contrast, due to the absence of misalignment and off-center loading, the vibration response in the non-dominant direction (z-direction) is extremely weak. Figure 12 (b)), the peak value is only Magnitude.

[0116] Under slight misalignment conditions ( Under these conditions, the overall response mode undergoes a significant evolution. In the primary load-bearing direction (y-direction), Figure 12 (c) The dominant high-frequency resonant band around 1380Hz remains stable, indicating that slight off-center loading does not significantly change the intrinsic high-frequency contact frequency of the gear teeth. However, in the non-dominant direction (z direction), Figure 12The change in (d) is particularly significant: the harmonic branches related to tooth surface meshing are strongly activated, and the peak response is more than two orders of magnitude higher than that in the centering condition, reaching [a certain level]. Meanwhile, a new resonant frequency band appears in the low-frequency range, and its characteristic frequency shifts slightly upward with the increase of rotational speed, evolving to about 244Hz at the highest speed, which reflects the regulating effect of rotational speed on the low-order support-rotor coupling natural frequency.

[0117] In moderate misalignment working conditions ( Under these conditions, the response characteristics undergo a more profound redistribution. In the y-direction ( Figure 12 (e)), the strongest resonant response energy is generated by the fundamental frequency branch. Transferred to and The branching indicates that misalignment alters the system's dynamic sensitivity to various meshing harmonics. Simultaneously, the low-frequency resonant band in the z-direction ( Figure 12 (f) It migrates to higher frequency bands, with its characteristic frequency reaching approximately 405 Hz at the highest rotational speed. This indicates that the change in misalignment has a much greater impact on the lower-order natural frequencies than simple rotational speed-induced modulation.

[0118] Under extremely severe misalignment conditions ( Under these conditions, the reconstruction of the system's dynamic characteristics is more drastic. In the y-direction ( Figure 12 (g) On the high-frequency resonance around 1380Hz, it remains the dominant mode, but This has become the primary source of motivation. However, in the non-principal direction (z direction), Figure 12 (h) Its low-frequency resonance frequency jumps sharply to about 766Hz, showing obvious characteristics of resonance amplification and transverse vibration energy transfer.

[0119] Therefore, it can be concluded that the system has two sets of modes with different evolutionary mechanisms: the first set is located at 1380Hz. The strong high-frequency resonant band in the vicinity remains basically constant under different degrees of misalignment and is not sensitive to off-center load; the second group is the support-rotor coupled resonant band located in the low-frequency band, whose characteristic frequency is mainly controlled by the degree of misalignment and off-center load, and it migrates significantly to the high-frequency band as the amount of misalignment increases.

[0120] Figure 13 A box plot of the statistical distribution of the system's variable-speed vibration response under the same operating conditions is presented, further describing the discrete law of response fluctuation with speed variation from a statistical perspective. In the main load direction (y-direction), under the centering condition (… Under these conditions, the response fluctuations exhibit interval-like variations, with a concentrated distribution in the medium-speed region. In the medium-to-high-speed region, due to the passage through the high-frequency resonance band, the interquartile range and whisker range significantly increase. As the misalignment increases to... and The response distribution tends to be more dispersed, and the fluctuation variability spreads to a wider continuous speed range. In extreme asymmetry (… Under these conditions, the main direction response fluctuations intensify, and it continues to oscillate in a high-energy mode over an extremely wide speed range, indicating that the misalignment severely reconstructs the response's sensitivity to speed.

[0121] In the non-dominant direction (z-direction), the response is highly concentrated within extremely low amplitudes when centered, with only weak activation at local speed points. Once misalignment is introduced, the interquartile range and extreme fluctuations in the low to medium speed range both increase significantly; at extreme misalignment ( Under these conditions, the response distribution expands dramatically, exhibiting a very large extreme value span and response skewness.

[0122] In summary, the dynamic vibration reduction and adjustment effects of spur gear pair tooth profile drum-shaped modification are not fixed, but strongly depend on the dynamic evolution of system stiffness and support state caused by bearing misalignment. The method of this invention can accurately and efficiently reproduce this fully coupled and complex dynamic evolution characteristics.

[0123] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A method for predicting vibration in a bearing-misaligned modified gear rotor system, characterized in that, Includes the following steps: Based on the obtained misalignment angle of the bearing and the tooth profile modification parameters of the gear, calculate the total tooth profile deviation of each slice after discrete slicing along the tooth width of the gear pair; Based on the total tooth profile deviation, a dynamic time-varying meshing stiffness model of the gear pair is established, including the continuous shear force coupling between the slices and the three-dimensional topological torsional compliance caused by non-uniform off-center loading. A nonlinear bearing support model is established to obtain the state-dependent dynamic bearing support stiffness that reflects the changes in the contact state of the rolling elements. A rotor dynamics model is established, and the dynamic time-varying meshing stiffness model and bearing dynamic support stiffness are coupled with the rotor dynamics model to construct a global dynamics model of the gear-rotor-bearing system. The global dynamic model is iteratively integrated in the time domain. Within each integration step, the relative misalignment state of the gear pair, the time-varying meshing stiffness, and the dynamic support stiffness of the bearing are updated in real time based on the instantaneous vibration response of the system to obtain the time-domain dynamic response of the system. The vibration characteristics of the system are then predicted based on the time-domain dynamic response.

2. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 1, characterized in that, Based on the obtained bearing misalignment angle and gear tooth profile modification parameters, the total tooth profile deviation of each slice after discrete slicing along the tooth width of the gear pair is calculated. The specific steps include: The gear teeth of the gear pair are uniformly discretized into multiple slices along the tooth width; Based on the misalignment angle of the bearings at the installation positions of the driving and driven gears, calculate the local tooth profile deviation of each slice caused by misalignment. The total tooth profile deviation of each slice is synthesized by combining the local geometric deviations of the slices caused by the tooth profile modification of the gear.

3. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 2, characterized in that, The tooth profile modification is described using a circular arc curve; The tooth profile modification is a circular arc-shaped modification symmetrical about the midpoint of the tooth width; The local shaping amount of each slice is determined based on the circular arc geometry equation by the radius of the drum-shaped arc and the end shaping amount; The local contact tilt angle of each slice is determined by geometrically coupling and superimposing the relative misalignment angle of the slice with the slope angle of the tangent line at the midpoint of the slice on the circular arc shaping curve.

4. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 1, characterized in that, The initial combined contact stiffness of each slice before coupling was determined in the following way: Introducing the equivalent cantilever stress state of each slice, the bending, shearing and axial compression multidimensional compliance components of each slice under meshing axial load are obtained; The deformation compliance of the tooth root matrix and the local Hertzian contact compliance of the tooth surface are coupled, and the multidimensional compliance components are superimposed in series. The initial comprehensive contact stiffness of each slice under local contact deformation is determined by reciprocal transformation.

5. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 1, characterized in that, In the dynamic time-varying meshing stiffness model, determining the continuous shear force coupling between the slices includes: Establish local interaction relationships between adjacent slices to characterize the shear constraints of the gear tooth entity, and utilize the slice coupling coefficient. Calculate the coupling stiffness between the i-th slice and the (i+1)-th slice. : In the formula, Let be the coupling stiffness between the i-th slice and the (i+1)-th slice. The slice coupling coefficient, For the i-th slice, the initial integrated contact stiffness before coupling is not considered. The initial composite contact stiffness of the (i+1)th slice before coupling is considered, where m is the gear module. Let be the slice thickness, and i be the slice number. , where n is the total number of slices; The coupling stiffness is assembled into the slice stiffness matrix of the contact control equation, and deformation coordination iteration including the load redistribution effect between slices is performed.

6. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 1, characterized in that, In the dynamic time-varying meshing stiffness model, the three-dimensional topological torsional compliance caused by non-uniform eccentric loading is determined, including: Based on the non-uniform load distribution on each slice, the additional torque generated by the normal force of each slice around the mid-surface of the gear tooth in two orthogonal directions is calculated respectively. The additional torsional strain energy of the gear teeth due to non-uniform eccentric loading in two orthogonal directions is calculated using the deformation energy method, and the topological torsional stiffness of the gear teeth in the two orthogonal directions is determined, which is used to characterize the three-dimensional topological torsional compliance of the gear teeth.

7. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 6, characterized in that, In the dynamic time-varying meshing stiffness model, the topological torsional stiffness in two orthogonal directions is synthesized along the meshing direction to obtain the total torsional stiffness. ; The equivalent contact stiffness of each slice is calculated and then superimposed in parallel along the tooth width direction. The total equivalent contact stiffness of the superimposed slices is then compared with the total torsional stiffness. By performing series elastic superposition, the overall time-varying meshing stiffness of the gear pair is finally synthesized. The calculation formula is as follows: In the formula, To determine the equivalent contact stiffness of the i-th slice after considering the slice coupling effect, The meshing force borne by the i-th slice, Let i be the total deformation of the i-th slice. For the i-th slice deformation component related to the driving pinion, For the i-th slice deformation component related to the driven large gear, Let be the Hertzian contact deformation component of the i-th slice. Let be the combined time-varying meshing stiffness of the gear pair, n be the total number of slices, and i be the slice number. , Let be the total torsional stiffness.

8. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 1, characterized in that, The nonlinear bearing support model is a quasi-static deep groove ball bearing model. Based on the geometric deformation coordination relationship between the rolling elements and the inner and outer raceways, and considering the three-dimensional force and torque balance relationship of rolling element friction, centrifugal force, and gyroscopic torque, the nonlinear dynamic support force vector acting on the inner ring of the bearing is calculated, and the displacement partial derivative is obtained to obtain the state-related dynamic support stiffness matrix.

9. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 1, characterized in that, A rotor dynamics model was established, including discretizing the shaft segment using Timoshenko beam elements and establishing the mass matrix, stiffness matrix, and gyroscope matrix of the shaft segment. The global dynamics model is represented by the following matrix-form control equations: In the formula, The global quality matrix of the system. Here is the global damping matrix of the system. This is the system's global gyroscope matrix. Let be the global stiffness matrix of the system, where the global stiffness matrix is... Includes the dynamic support stiffness of the bearing and the combined time-varying meshing stiffness of the gear pair. and rotor shaft stiffness, Let be the generalized displacement vector of the system. Let be the generalized velocity vector of the system, which is The first derivative with respect to time, Let be the generalized acceleration vector of the system, which is The second derivative with respect to time, It is a generalized excitation vector.

10. The vibration prediction method for a bearing misalignment modified gear rotor system according to claim 1, characterized in that, The global dynamics model is subjected to stepwise iterative integration in the time domain, and a double-nested time-domain numerical integration is performed using the Newmark-β integration algorithm combined with the Newton-Raphson iterative algorithm. Within each integration time step, the relative misalignment angle of the gear pair under the current step is calculated in real time using the instantaneous translational and rotational responses of the gear nodes. Based on this, the time-varying meshing stiffness and bearing dynamic support stiffness are recalculated and the dynamic state is updated until the dynamic response of the current step meets the convergence condition, and then the next integration time step is entered. The vibration characteristics of the system are predicted based on the time-domain dynamic response, including: The initial transient response in the time-domain dynamic response is removed, and the steady-state vibration response is extracted and subjected to time-domain waveform analysis, frequency-domain spectrum analysis, and response fluctuation statistical analysis to predict the dynamic vibration characteristics of the system under different misalignment and speed conditions.

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