System dynamics modeling method and system for meshless flexible gear pair

Abstract

The invention relates to the technical field of gear modeling, and discloses a system dynamics modeling method and system for a meshless flexible gear pair, and the method comprises the steps: constructing a flexible gear rotor system coordinate system, calculating a displacement vector of any point of a gear when the system moves, carrying out the time derivation of the displacement vector, and obtaining a deformation velocity vector; calculating the total kinetic energy of the gear rotor system based on the deformation velocity vector; calculating positive strain and shear strain potential energy of the small and large gears according to the displacement vector, obtaining matrix boundary condition potential energy by combining the matrix rigidity, calculating gear tooth meshing elastic potential energy according to the meshing rigidity and the relative position vector, and summing various types of potential energy to obtain total potential energy of the system. Finally, the total kinetic energy and the potential energy are substituted into the analysis mechanics principle, a system dynamics partial differential control equation set is deduced through variational operation, and meshless flexible gear pair system dynamics modeling is completed. According to the method, the problems of traditional finite element grid division and large-scale calculation are avoided, and the modeling efficiency and the complex working condition adaptability are improved.

Description

Technical Field

[0001] This invention relates to the field of gear modeling technology, and specifically to a system dynamics modeling method and system for a meshless flexible gear pair. Background Technology

[0002] With the continuous upgrading of advanced mechanical design requirements, lightweight gear transmission systems have become one of the core technologies in the field of high-performance mechanical transmission, and are accelerating their development towards higher power density, more precise transmission accuracy, better noise control, and stronger environmental adaptability. Given the inherent high flexibility and high sensitivity of lightweight structures, research in this field needs to focus on ensuring the stability and reliability of the system under extreme loads and high-speed conditions to meet the requirements of complex and ever-changing operating environments.

[0003] During the operation of thin-walled gears, the transmission system is subjected to the coupling effect of internal and external excitation forces, making rotor system vibration a significant issue and posing a severe challenge to its safe and stable operation. This is especially true in helical gear transmission systems, where the increased flexibility of thin-walled gears makes them prone to significant vibration during operation; simultaneously, the additional axial excitation introduced by the helical gears further exacerbates the risk of system instability. In addition to meshing-induced vibration, flexible gears undergo flexible deformation at high speeds due to inertial forces, centrifugal forces, and gyroscopic effects; coupled with the stringent requirements for axial force balance in gear operation, the system's sensitivity to dynamic loads is significantly increased. Furthermore, modern high-performance mechanical equipment places even more stringent demands on the performance indicators of gear systems (such as higher speeds, greater transmission torque, and longer service life), which urgently necessitates further in-depth research on the gyroscopic effect and high-frequency resonance phenomena under high-speed rotation.

[0004] Currently, in the field of flexible component modeling, most methods employ finite element method (FEM) modeling for gear components. This method requires fine mesh generation and large-scale matrix calculations, resulting in excessive computational resource consumption during the modeling process, which to some extent restricts modeling efficiency and the progress of subsequent analysis. Summary of the Invention

[0005] In view of this, in order to solve the problems of high computational cost and low efficiency of existing gear dynamics modeling methods, this invention provides a system dynamics modeling method and system for meshless flexible gear pairs, which can realize meshless finite element modeling and simplify the calculation process of flexible component deformation.

[0006] In a first aspect, the present invention provides a system dynamics modeling method for meshless flexible gear pairs, comprising:

[0007] Establish a coordinate system for the flexible gear rotor system, and calculate the displacement vector of any point of the gear in the coordinate system when the gear rotor system moves;

[0008] The displacement vector is differentiated over time to obtain the corresponding deformation velocity vector, and the total kinetic energy of the gear rotor system is calculated based on the deformation velocity vector.

[0009] The normal strain potential energy and shear strain potential energy corresponding to the pinion and gear are calculated based on the displacement vector. The base boundary condition potential energy corresponding to the pinion and gear is calculated based on the displacement vector and the base stiffness. The elastic potential energy formed by the meshing of the gear teeth is calculated based on the meshing stiffness and relative position vector of the pinion and gear. The normal strain potential energy, the shear strain potential energy, the base boundary condition potential energy and the elastic potential energy are summed to obtain the total potential energy of the gear rotor system.

[0010] Substituting the total kinetic energy and total potential energy of the system into the principles of analytical mechanics, the partial differential control equations describing the system dynamics are derived through variational operations, thus completing the system dynamics modeling of the meshless flexible gear pair.

[0011] The system dynamics modeling method for meshless flexible gear pairs provided in this invention establishes a system dynamics model for meshless flexible gear pairs through a complete process of coordinate system construction, kinetic and potential energy calculation, and dynamic modeling derivation. Its core advantages lie in: Systematic modeling logic: Starting from the basic coordinate system and displacement vectors, it derives kinetic energy, potential energy, and control equations layer by layer, constructing a dynamic analysis system covering all elements of gear motion (displacement, velocity, strain, meshing, etc.), overcoming the limitations of traditional modeling which relies heavily on simplification assumptions and struggles to accurately characterize flexible properties; Accuracy of multi-physical quantity coupling: Simultaneously considering potential energy in multiple dimensions such as normal strain, shear strain, matrix boundary, and gear meshing, as well as kinetic energy related to micro-displacement and rotational speed, it accurately recreates the mechanical response of flexible gears under complex working conditions, providing high-precision theoretical support for vibration control and reliability design of gear systems in high-speed, heavy-load scenarios; Adaptability to meshless modeling: The derivation process conforms to the requirements of meshless methods, avoiding the difficulties of traditional finite element mesh generation and large-scale calculations, improving modeling efficiency and adaptability to complex working conditions, and assisting in the innovative research and development of high-performance mechanical transmission systems (such as aero-engine transmission chains).

[0012] In one optional implementation, establishing the coordinate system of the flexible gear rotor system includes: for any point on the flexible gear during system operation, establishing a local coordinate system within the cross-section where the point is located, with the central axis of the cross-section as the origin, and representing the arbitrary point in the local coordinate system as coordinates [x...]. i 0z i ], where x i and z i The macroscopic position coordinates of any point in the local coordinate system with the central axis as the origin;

[0013] Establish a fixed coordinate system, with the center of the flexible gear as the origin of the fixed coordinate system. Transform any point from the local coordinate system to the fixed coordinate system to obtain the coordinates r of the arbitrary point in the fixed coordinate system. 0i =[R i +x i 0z i ], where R i This represents the diameter of the central shaft of the i-th gear.

[0014] This invention achieves the following key benefits through step-by-step construction and coordinate transformation between a local coordinate system and a fixed coordinate system: Accurate position representation: Using the cross-sectional central axis and the gear center as dual origins, the local macroscopic position and global absolute position of any point on the gear are accurately defined, solving the problem of complex point motion trajectories during flexible gear deformation and the difficulty in adapting to traditional coordinate systems, providing a reliable coordinate benchmark for subsequent displacement and strain calculations; Multi-scale analysis compatibility: The local coordinate system focuses on the microscopic deformation of the gear (such as local flexible deformation of the tooth surface), while the fixed coordinate system supports system-level macroscopic motion (such as the overall rotation of the gear rotor), balancing microscopic and macroscopic analysis needs, allowing the model to capture both subtle tooth surface strain and correlate with the global dynamic behavior of the system; Convenient parametric modeling: Introducing the gear central axis diameter as a coordinate transformation parameter adapts to gear designs with different modules and sizes, enhancing the method's versatility for serialized gear transmission systems (such as multi-gear gearboxes), and reducing the repetitive development costs of modeling multi-specification gears.

[0015] In one optional implementation, the displacement vector is obtained by superimposing the coordinates of any point on the gear with the corresponding micro-movement, wherein the micro-movement includes micro-movement transverse displacement in the radial, tangential and axial directions, and micro-movement rotation around the radial, tangential and axial coordinate axes.

[0016] The expression for the deformation velocity vector includes the gear rotation speed, the partial derivative of the fretting displacement with respect to time, and the partial derivative with respect to the rotation angle.

[0017] This invention, through superimposing the coordinates of any point on the gear with the micro-movements encompassing radial, tangential, and axial directions, as well as the corresponding micro-rotation angles of the shaft system, fully captures the subtle deformations of the flexible gear's six degrees of freedom. This overcomes the dynamic response distortion problem caused by the simplification of motion forms in traditional models, and accurately depicts the complex motion state of gears operating at high speeds. Simultaneously, the deformation velocity vector incorporates the time and angle partial derivatives of the rotational speed and micro-movements, effectively coupling the dynamic characteristics of rigid body rotation and flexible deformation. This accurately reflects the influence of gyroscopic effects and speed changes on the system's kinetic energy, laying the foundation for subsequent accurate calculation of the gear-rotor system's dynamic parameters and improving the model's adaptability and analytical accuracy in high-speed, highly flexible gear transmission scenarios.

[0018] In one optional implementation, the total kinetic energy of the system is the sum of the kinetic energies of the pinion and gear, and the formula for calculating the kinetic energy of a single gear is:

[0019]

[0020] Among them, K i Let R represent the kinetic energy of the i-th gear, where i=1 represents the pinion, i=2 represents the gear, ρ is the density of the gear components, and R is the kinetic energy of the gear. i x represents the diameter of the central shaft of the i-th gear. i This represents the polar coordinate distance from any point on the i-th gear to the central axis. Let A represent the deformation velocity vector of any point on the i-th gear in the coordinate system, A represent the cross-sectional area of ​​the gear, and θ represent the angular integral domain of the gear.

[0021] This invention decomposes the total kinetic energy of the system into the sum of the kinetic energies of the pinion and gear, accurately adapting to the differentiated motion characteristics of the pinion under high speed and light load, and the gear under low speed and heavy load in gear transmission. This avoids the distortion of kinetic energy distribution caused by traditional simplified calculations, and clearly quantifies the kinetic energy ratio of the two gears under different working conditions, providing accurate data support for the energy flow analysis of the transmission system. At the same time, the formula parameters are directly related to the gear design and manufacturing parameters, which not only improves the accuracy of kinetic energy calculation, but also enhances the versatility of the model in gear pairs of different specifications and materials. This provides a reliable kinetic energy foundation for subsequent system total potential energy coupling analysis and dynamic equation derivation, and assists in the vibration control and reliability design of high-speed, highly flexible gear transmission systems.

[0022] In one optional implementation, the normal strain potential energy and shear strain potential energy corresponding to the pinion and gear respectively are calculated based on the displacement vector, including:

[0023] The normal strain and shear strain are derived based on the displacement vector;

[0024] The normal strain potential energy corresponding to the pinion and gear is calculated based on the following formula:

[0025]

[0026] The shear potential energy corresponding to the pinion and gear is calculated based on the following formula:

[0027]

[0028] Where i=1 represents the small gear, i=2 represents the large gear, and R i x represents the diameter of the central shaft of the i-th gear. i This represents the polar coordinate distance from any point on the i-th gear to the central axis. Let represent the kinetic energy vector of any point on the i-th gear in a fixed coordinate system, A represent the cross-sectional area of ​​the gear, θ represent the angular integration domain of the gear, and σ represent the kinetic energy vector of any point on the i-th gear. i Let σ represent the normal stress amplitude of the i-th gear component and σi represent the normal stress amplitude of the i-th gear component. i =Eε i , ε i τ is the normal strain of the i-th gear component, and E is the Young's modulus of the material; rt,i ,τ tz,i Let τ represent the shear stress in the radial-tangential plane and the shear stress in the tangential-axial plane of the i-th gear component, and let τ i =μGγ i Where μ is the shear correction factor, G is the shear modulus, and γ rt,i ,γ tz,i Let be the shear strain in the radial-tangential plane and the shear strain in the tangential-axial plane.

[0029] This invention derives normal strain and shear strain based on displacement vectors, ensuring that strain calculations are directly related to the actual deformation state of the gear. This avoids deviations between traditional empirical strain assumptions and real-world conditions, providing accurate mechanical input for subsequent potential energy calculations. Furthermore, it calculates the two strain potential energies separately using differentiated formulas, incorporating geometric parameters such as the gear's central shaft diameter and polar coordinate distance, as well as material properties such as Young's modulus E, shear modulus G, and shear correction coefficient. This allows for precise quantification of the strain potential energy differences between the pinion and gear due to size and material variations, while also comprehensively covering the influence of shear strain in the radial-tangential and tangential-axial planes of the gear. This overcomes the problem of incomplete mechanical characterization caused by traditional single-consideration of normal strain potential energy. Simultaneously, by combining cross-sectional area and angular integral domain integration, it can precisely capture the strain potential energy distribution in different regions of the gear (such as the tooth root and tooth surface), providing high-precision potential energy data support for subsequent calculations of the total system potential energy and derivation of dynamic equations. This effectively improves the accuracy of the model's analysis of gear deformation, vibration, and other dynamic behaviors, contributing to the structural optimization and reliability design of high-performance flexible gear pairs.

[0030] In one optional implementation, when calculating the normal strain based on the displacement vector, the components of the displacement vector at any point on the gear along the radial, tangential, and axial directions are Taylor expanded, and small quantities of a preset order or higher are ignored to obtain an expression, wherein the normal strain is related to the displacement gradient in the radial direction, the displacement gradient in the tangential direction, and the displacement change in the axial direction at that point.

[0031] When calculating shear strain, only the shear strain in the radial-tangential plane and the shear strain in the tangential-axial plane are considered. The shear strain in the radial-tangential plane is composed of the sum of the partial derivatives of the radial displacement with respect to the tangential coordinate and the partial derivatives of the tangential displacement with respect to the radial coordinate. The shear strain in the tangential-axial plane is composed of the sum of the partial derivatives of the tangential displacement with respect to the axial coordinate and the partial derivatives of the axial displacement with respect to the tangential coordinate.

[0032] This invention employs Taylor expansion of displacement vector components and ignores higher-order minor quantities. While ensuring computational accuracy, it simplifies the normal strain calculation model, solving the problem of computational explosion and difficulty in engineering application caused by full-order expansion. This allows high-precision strain analysis to be completed with conventional computing resources, clearly revealing the relationship between normal strain and radial / tangential displacement gradients, axial displacement changes, and the relationship between shear strain and the sum of partial derivatives of displacement in different planes. It clearly reveals the transformation law of displacement-strain during gear deformation, making it easier for engineers to understand and adjust model parameters (such as reducing strain in key areas by optimizing tooth profile curves). It focuses on shear strain in the "radial-tangential, tangential-axial" planes, closely matching the actual stress characteristics of gears (such as in-plane shearing on the tooth surface during meshing), avoiding the introduction of irrelevant strain components that interfere with the calculation, and improving the model's relevance and effectiveness for engineering problems (such as tooth surface scuffing failure analysis).

[0033] In one alternative implementation, the potential energy of the base boundary conditions corresponding to the pinion and the gear is calculated based on the following formula:

[0034]

[0035] Where, k ri ,k ti ,k ai Let u be the base stiffness of the i-th gear component. i ,v i ,w i Let represent the radial, tangential, and axial micro-displacements of the i-th gear component, respectively, and let D represent the circumferential integral domain of the entire gear, where D = [0, 2π).

[0036] This invention couples the matrix stiffness with the gear's fretting displacement, quantifying the constraint effect of the matrix on gear deformation (such as the influence of the connection stiffness between the gear and the shaft / box), solving the problem of traditional models "ignoring matrix constraints, leading to idealized boundary conditions," and improving the fit between system dynamics analysis and actual working conditions (such as gearbox gear-shaft coupling vibration analysis). The formulas cover the constraint potential energy of radial, tangential, and axial fretting displacements, supporting modeling of different matrix constraint forms such as "interference fit, elastic support, and sliding bearings," providing a unified analysis framework for diverse transmission systems. By adjusting the matrix stiffness parameters, the influence of different connection processes and support structures on gear dynamic characteristics can be simulated, providing a quantitative basis for integrated "gear-matrix" design (such as lightweight gearbox topology optimization), and contributing to the overall performance improvement of the transmission system.

[0037] In one alternative implementation, the elastic potential energy generated by tooth meshing is calculated based on the following formula:

[0038]

[0039] Among them, P m k represents the potential energy of the bearing boundary conditions. m β represents the meshing stiffness between the large gear and the small gear, β represents the helix angle between the large gear and the small gear, v1 is the tangential elastic deformation of the small gear at the meshing position, v2 is the tangential elastic deformation of the large gear at the meshing position, w1 is the axial elastic deformation of the small gear at the meshing position, and w2 is the axial elastic deformation of the large gear at the meshing position.

[0040] The embodiments of this invention include the bearing boundary condition potential energy in the elastic potential energy calculation formula, supporting multi-potential energy coupling calculation of gear meshing and bearing support, and completely restoring the mechanical transmission path of the transmission system (such as the vibration transmission of meshing force to the housing through the bearing), providing an accurate model for vibration and noise tracing and vibration isolation design of the transmission system; meshing stiffness, as a key parameter, can be associated with failure processes such as gear wear and tooth surface contact fatigue (such as wear leading to a decrease), assisting in the dynamic analysis of the entire life cycle of the gear.

[0041] In one optional implementation, the analytical mechanics principle is Hamilton's principle, by substituting the system's total kinetic energy K and total potential energy U into the variational expression of Hamilton's principle. The partial differential equations for each degree of freedom of the meshless gear dynamics were obtained through calculation.

[0042] This invention relies on the principles of classical analytical mechanics to provide a rigorous mathematical foundation for dynamic modeling, ensuring that the derived partial differential equations satisfy physical constraints such as energy conservation and variational extrema, thus solving the problems of "empirical modeling lacking theoretical support and questionable reliability of results." By directly deriving multi-degree-of-freedom (e.g., 6 degrees of freedom for micro-displacement) partial differential equations through variational operations, it efficiently handles multi-degree-of-freedom coupled vibration problems of flexible gears, avoiding the error accumulation of traditional "single-degree-of-freedom simplification - multi-degree-of-freedom extension" methods, and improving the model's analytical accuracy for complex dynamic behaviors (e.g., modal coupling, parametric resonance). The weak form of Hamilton's principle is naturally compatible with meshless methods (e.g., radial basis function method). The control equations derived through variational derivation can be directly solved using meshless discretization, avoiding the finite element mesh dependency problem, and providing an efficient and accurate means for dynamic analysis of flexible gears under extreme conditions (e.g., transient impact, large deformation).

[0043] Secondly, the present invention provides a system dynamics modeling system for meshless flexible gear pairs, the system comprising:

[0044] The coordinate establishment module is used to establish the coordinate system of the flexible gear rotor system and calculate the displacement vector of any point of the gear in the coordinate system when the gear rotor system moves.

[0045] The system total kinetic energy calculation module is used to perform time derivative on the displacement vector to obtain the corresponding deformation velocity vector, and calculate the total kinetic energy of the system based on the deformation velocity vector;

[0046] The system's total potential energy calculation module is used to calculate the normal strain potential energy and shear strain potential energy corresponding to the pinion and gear respectively based on the displacement vector, calculate the base boundary condition potential energy corresponding to the pinion and gear respectively based on the displacement vector and base stiffness, and calculate the elastic potential energy formed by gear meshing based on the meshing stiffness and relative position vector of the pinion and gear, and sum the various potential energies to obtain the total potential energy of the system.

[0047] The dynamics modeling module substitutes the total kinetic energy and total potential energy of the system into the principles of analytical mechanics, and derives the partial differential control equations describing the system dynamics through variational operations, thus completing the system dynamics modeling of the meshless flexible gear pair.

[0048] Thirdly, the present invention provides a computer device, comprising: a memory and a processor, wherein the memory and the processor are communicatively connected to each other, the memory stores computer instructions, and the processor executes the computer instructions to perform the system dynamics modeling method for meshless flexible gear pairs described in the first aspect or any corresponding embodiment thereof.

[0049] Fourthly, the present invention provides a computer-readable storage medium storing computer instructions for causing a computer to execute the system dynamics modeling method for a meshless flexible gear pair described in the first aspect or any corresponding embodiment thereof.

[0050] Fifthly, the present invention provides a computer program product, including computer instructions for causing a computer to execute the system dynamics modeling method for a meshless flexible gear pair described in the first aspect or any corresponding embodiment thereof. Attached Figure Description

[0051] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0052] Figure 1 This is a schematic flowchart of a system dynamics modeling method for a meshless flexible gear pair according to an embodiment of the present invention;

[0053] Figure 2 (a) is a top view of the flexible gear meshing state. Figure 2 (b) is a front view of a cross-section of a single flexible gear (including the coordinates of any point). Figure 2 (c) is a side view of the meshing interface of the flexible gear pair;

[0054] Figure 3 This is a schematic diagram of key steps in the modeling method for a meshless flexible gear pair according to an embodiment of the present invention;

[0055] Figure 4 This is a structural block diagram of a system dynamics modeling system for a meshless flexible gear pair according to an embodiment of the present invention;

[0056] Figure 5 A schematic diagram of the hardware structure of a computer device according to an embodiment of the present invention. Detailed Implementation

[0057] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0058] Existing flexible component modeling methods employ finite element method (FEM) modeling of gear components, requiring 3D modeling, mesh generation, and large-scale matrix calculations. This results in high repetitive computation costs, long computation times, and low efficiency for large-scale calculations. To address these shortcomings, this embodiment provides a meshless flexible gear pair system dynamics modeling method. It uses the elastic component energy method to derive the deformation velocity vector, stress, and strain of the gear system under rotational conditions. Considering the flexible deformation of the flexible gear transmission system, a unified coordinate system is established for the large and small gears. The basic idea for dynamic modeling of a fully flexible gear rotor system is to treat the flexible deformation of the gear rotor system as the motion of countless random particles. By providing general expressions for the velocity vector, normal strain, shear strain, and boundary condition constraints of all random particles, the kinetic and potential energy expressions of the entire gear system can be calculated through global integration.

[0059] Figure 1 This is a flowchart of a system dynamics modeling method for a meshless flexible gear pair according to an embodiment of the present invention. Figure 1 As shown, the process includes the following steps:

[0060] Step S1: Establish the coordinate system of the flexible gear rotor system, and calculate the displacement vector of any point of the gear in the coordinate system when the gear rotor system moves.

[0061] Specifically, Figure 2 Image (a) is a top view of the flexible gears in meshing state. Figure 2 Image (b) is a front view of a cross-section of a single flexible gear (including the coordinates of any point). Figure 2 Image (c) is a side view of the meshing interface of the flexible gear pair. For example... Figure 2 As shown in (b), during system operation, any point on the flexible gear can be represented by coordinates [x] in the cross-section where that point is located. i 0z i The form of ], where x i and z i These are the coordinates of the macroscopic position of any point within a local coordinate system established with the central axis as the origin. By placing this point in a fixed coordinate system and using the center of the gear as the origin, we can obtain the coordinates of any point within the cross-section. Figure 2 (a) has coordinates r 0i =[R i +x i 0z i The subscript i (i = 1 or 2) indicates that the corresponding variable is a small gear and a large gear, respectively, where R i This represents the diameter of the central shaft of the i-th gear.

[0062] The local coordinate system established in this invention focuses on the microscopic deformation of the gear (such as the local flexible deformation of the tooth surface), while the fixed coordinate system supports the macroscopic motion of the system (such as the overall rotation of the gear rotor). This takes into account both microscopic and macroscopic analysis needs, allowing the model to capture both the subtle strain of the tooth surface and the global dynamic behavior of the system. The diameter of the gear's central shaft is introduced as a coordinate transformation parameter to adapt to gear designs with different modules and sizes, enhancing the method's versatility for serialized gear transmission systems (such as multi-gear gearboxes) and reducing the repetitive development costs of modeling multi-specification gears.

[0063] Furthermore, the Timoshenko beam element (a mechanical model element used to describe the micro-displacement of a flexible gear rotor system, whose core feature is that it considers the shear deformation and rotational inertia of the beam, and can accurately reflect the complex deformation state of the flexible component during motion) has 6 degrees of freedom of micro-displacement, where u, v, and w are the micro-lateral displacements in the radial, tangential, and axial directions, respectively, and φ x , φ y , φ z These represent the micro-rotation angles about the radial, tangential, and axial coordinate axes, respectively. During system operation, micro-displacements will exist. Superimposing the macroscopic coordinates with the corresponding micro-displacements yields the displacement vector of any point at a specific moment:

[0064]

[0065] Among them, e r ,e t and e a These represent the unit vectors parallel to the radial, tangential, and axial coordinate directions, respectively. The radial and tangential unit vectors e at any point during motion... r ,e t Both will change over time. The subscript i (i = 1 or 2) indicates that the corresponding variable is a small gear and a large gear, respectively.

[0066] This invention achieves significant advantages by clearly defining "the displacement vector includes all elements of micro-motion displacement and the velocity vector is associated with multi-dimensional partial derivatives": Completeness of motion description: Micro-motion displacement is subdivided into 6 degrees of freedom, namely "lateral movement + rotation", which fully covers the radial contraction, tangential torsion, axial movement and oscillation around the axis during the deformation of flexible gears, solving the problem of "ignoring some micro-motion forms and causing distortion of dynamic response" in traditional models (such as axial oscillation caused by axial force of helical gears); Dynamic coupling characterization: The expression of the velocity vector incorporates "rotation speed, micro-motion displacement time / rotation angle partial derivatives", accurately capturing the coupling effect of "gear fixed-axis rotation + flexible deformation vibration" (such as the mutual influence of gyro effect and tooth surface vibration under high-speed rotation), providing key dynamic input for vibration and noise control of high-speed, highly flexible gear systems (such as electric drive gearboxes of new energy vehicles).

[0067] Step S2: Perform time derivative on the displacement vector to obtain the corresponding deformation velocity vector, and calculate the total kinetic energy of the system based on the deformation velocity vector.

[0068] Specifically, by differentiating the displacement vector over time, the deformation velocity vector at any point can be obtained. Specifically:

[0069]

[0070] in, Such superscript marks indicate that the partial derivative of the small displacement with respect to time is taken (the same applies to the small rotation angle). This type of upper right punctuation indicates that the partial derivative of the small displacement with respect to the rotation angle is taken (the same applies to the small rotation angle), Ω i This indicates the rotational speed of the corresponding pinion or gear.

[0071] The total kinetic energy of the system is the sum of the kinetic energies of the pinion and gear, K = K1 + K2. The formula for calculating the kinetic energy of a single gear is:

[0072]

[0073] Among them, K i Let R represent the kinetic energy of the i-th gear, where i=1 represents the pinion, i=2 represents the gear, ρ is the density of the gear components, and R is the kinetic energy of the gear. i x represents the diameter of the central shaft of the i-th gear. i This represents the polar coordinate distance from any point on the i-th gear to the central axis. Let A represent the deformation velocity vector of any point on the i-th gear in the coordinate system, A represent the cross-sectional area of ​​the gear, and θ represent the angular integral domain of the gear.

[0074] This kinetic energy calculation formula is based on the deformation velocity vector and the gear's geometric and physical parameters (density, cross-sectional area, etc.). It derives the integral formula for the kinetic energy of a single gear, breaking through the limitations of the traditional simplification to rigid body rotational kinetic energy and ignoring flexible deformation kinetic energy. It accurately includes the additional kinetic energy caused by flexible deformation of the tooth surface (such as the vibration kinetic energy when a thin-walled gear rotates at high speed), thus improving the realism of system dynamic analysis.

[0075] Step S3: Calculate the normal strain potential energy and shear strain potential energy corresponding to the pinion and gear respectively based on the displacement vector; calculate the base boundary condition potential energy corresponding to the pinion and gear respectively based on the displacement vector and the base stiffness; calculate the elastic potential energy formed by gear meshing based on the meshing stiffness and relative position vector of the gear and gear; and sum the various potential energies to obtain the total potential energy of the system.

[0076] like Figure 3As shown, before calculating the normal strain potential energy and shear strain potential energy corresponding to the pinion and gear respectively in this embodiment of the invention, it is necessary to confirm the normal strain and shear strain. For the normal strain and shear strain of the pinion and gear, the linear deformation of the centerline is obtained using the material deformation theory as follows:

[0077]

[0078] Where θ represents the angular coordinates of any point, and dθ is the angular coordinates of a point near that point with the smallest angular difference, η i =u i ′+z i φ yi ′-v i +z i φ xi -x i φ zi ξ represents the radial component of gear deformation. i =v′ i -z i φ xi ′+x i φ zi ′+u i +z i φ yi This represents the tangential component of the gear deformation, where... This represents the axial component of the gear deformation. Based on the formulas for calculating deformation and strain, the strain is calculated as follows:

[0079]

[0080] in The length of the micro-arc segment before deformation. Let be the length of the deformed micro-arc segment. Expanding the above equation using Taylor and neglecting small quantities of order 3 and above, we obtain the expression for the normal strain as follows: By Taylor expansion of the displacement vector components and neglecting higher-order small quantities, the normal strain calculation model is simplified while ensuring calculation accuracy. This solves the problem of "full-order expansion leading to an explosion in computational load and difficulty in engineering applications", allowing high-precision strain analysis to be completed with conventional computing resources.

[0081] Based on the theory of material shear deformation, the shear strain in the radial-tangential plane and the shear strain in the tangential-axial plane can be calculated and expressed by the following formula:

[0082]

[0083] Where, γ rt,i γ is the shear strain in the radial-tangential plane. tz,iThe shear strain is in the tangential-axial plane. Since warping deformation is neglected, the shear strain in the radial-axial plane is 0, i.e., γ. rz =0.

[0084] This invention clarifies the relationship between normal strain and "radial / tangential displacement gradient and axial displacement change", and the relationship between shear strain and "the sum of partial derivatives of displacement in different planes". It clearly reveals the transformation law of "displacement-strain" during gear deformation, which is convenient for engineers to understand and adjust model parameters (such as reducing strain in key areas by optimizing tooth profile curves). It focuses on the shear strain in the "radial-tangential and tangential-axial" planes, which fits the actual stress characteristics of gears (such as the tooth surface being mainly under in-plane shear during meshing), avoids introducing irrelevant strain components to interfere with the calculation, and improves the model's pertinence and effectiveness for engineering problems (such as tooth surface scuffing failure analysis).

[0085] Furthermore, based on the displacement vectors and normal strain of the small and large gears, the method for calculating the normal strain potential energy corresponding to each gear is as follows:

[0086]

[0087] Based on the displacement vectors and shear strain of the small and large gears, the method for calculating the shear potential energy corresponding to each gear is as follows:

[0088]

[0089] Among them, K i Let R represent the kinetic energy of the i-th gear, where i=1 represents the pinion, i=2 represents the gear, ρ represents the density of the gear components, and R represents the kinetic energy of the gear. i x represents the diameter of the central shaft of the i-th gear. i This represents the polar coordinate distance from any point on the i-th gear to the central axis. Let represent the kinetic energy vector of any point on the i-th gear in a fixed coordinate system, A represent the cross-sectional area of ​​the gear, θ represent the angular integration domain of the gear, and σ represent the kinetic energy vector of any point on the i-th gear. i Let σ represent the normal stress amplitude of the i-th gear component and σi represent the normal stress amplitude of the i-th gear component. i =Eε i E is the Young's modulus of the material, ε i τ is the normal strain of the i-th gear component. rt,i ,τ tz,i Let τ represent the shear stress in the radial-tangential plane and the shear stress in the tangential-axial plane of the i-th gear component, and let τ i =μGγ i Where μ is the shear correction factor, G is the shear modulus, and γ rt,i ,γ tz,i Let be the shear strain in the radial-tangential plane and the shear strain in the tangential-axial plane.

[0090] This invention simultaneously considers the potential energy contributions of normal strain (tension / compression) and shear strain (in-plane shear), comprehensively covering the main mechanical forms of flexible gear deformation (such as tooth root bending normal strain and tooth surface contact shear strain). It solves the problem of traditional models being "dominated by a single strain energy and unable to reflect complex stress states" (such as multi-directional strain coupling during helical gear meshing). Based on integral calculations at any point on the gear, the strain potential energy distribution in stress concentration areas on the tooth surface (such as the tooth root transition fillet and the vicinity of the meshing line) can be located, providing a micromechanical basis for gear fatigue life prediction and tooth profile optimization design, thereby improving the gear's failure resistance and service reliability.

[0091] Furthermore, boundary condition constraints need to be applied to the gear teeth themselves. In this embodiment of the invention, based on the displacement vectors and base stiffness of the small and large gears, the potential energy of the base boundary conditions corresponding to each gear is calculated. The method is as follows:

[0092]

[0093] Where, k ri ,k ti ,k ai Let u be the base stiffness of the i-th gear component. i ,v i ,w i Let represent the radial, tangential, and axial fretting displacements of the i-th gear component, respectively, and D represent the circumferential integral domain of the entire gear, D = [0, 2π). The formula encompasses the constraint potential energy of radial, tangential, and axial fretting displacements, supporting modeling of different base constraint forms such as "interference fit, elastic support, and sliding bearing," providing a unified analysis framework for diverse transmission systems (such as high-precision constraints in aerospace transmission chains and heavy-load constraints in engineering machinery gearboxes). By adjusting the base stiffness parameters, the influence of different connection processes and support structures on gear dynamics can be simulated, providing a quantitative basis for integrated "gear-base" design (such as topology optimization of lightweight gearboxes), and contributing to the overall performance improvement of the transmission system.

[0094] The elastic potential energy generated by the meshing of the gear teeth is calculated based on the meshing stiffness and relative position vector of the large and small gears. The method is as follows:

[0095]

[0096] Among them, P m k represents the potential energy of the bearing boundary conditions. mThe formula represents the meshing stiffness between the large and small gears, β represents the helix angle of the large and small gears, v1 is the tangential elastic deformation of the small gear at the meshing position, v2 is the tangential elastic deformation of the large gear at the meshing position, w1 is the axial elastic deformation of the small gear at the meshing position, and w2 is the axial elastic deformation of the large gear at the meshing position. The formula includes the bearing boundary condition potential energy, supporting multi-potential energy coupling calculations of gear meshing and bearing support, fully restoring the mechanical transmission path of the transmission system (such as the vibration transmission of meshing force to the housing via the bearing), providing an accurate model for vibration and noise source tracing and vibration isolation design of the transmission system; meshing stiffness, as a key parameter, can be correlated with failure processes such as gear wear and tooth surface contact fatigue (such as wear leading to a decrease in k_m), assisting in the dynamic analysis of the gear's entire life cycle, supporting intelligent operation and maintenance (such as fault early warning based on changes in meshing potential energy) and service reliability assessment.

[0097] By summing the potential energy of the large gear and small gear under normal strain, shear strain, matrix boundary conditions, and elasticity, the total potential energy P of the gear-rotor system is obtained as follows:

[0098] P = P m +P f,1 +P shear,1 +P strain,1 +P f,2 +P shear,2 +P strain,2 (11)

[0099] Among them, P f,1 ,P shear,1 ,P strain,1 , representing the potential energy of the base boundary condition of the pinion, the shear strain potential energy, and the normal strain potential energy, respectively. P f,2 ,P shear,2 ,P strain,2 , respectively represent the potential energy of the base boundary condition of the large gear, the shear strain potential energy, and the normal strain potential energy.

[0100] Step S4: Substitute the total kinetic energy and total potential energy of the system into the principles of analytical mechanics, and derive the partial differential control equations describing the system dynamics through variational operations, thus completing the system dynamics modeling of the meshless flexible gear pair.

[0101] In this embodiment of the invention, the total kinetic energy and total potential energy are substituted into Hamilton's principle, and the integral of each kinetic and potential energy over time is set to 0, that is:

[0102]

[0103] Based on the principles of classical analytical mechanics, this method provides a rigorous mathematical foundation for dynamic modeling, ensuring that the derived partial differential equations satisfy physical constraints such as energy conservation and variational extrema, thus solving the problem of "empirical modeling lacking theoretical support and questionable reliability of results".

[0104] Further calculation of equation (12) yields the partial differential equations for each degree of freedom in the dynamics of the meshless gear:

[0105]

[0106]

[0107] The expressions for each coefficient are as follows:

[0108]

[0109] Where h i ,b i Let be the cross-sectional width and height of the i-th gear component, respectively. δ(ψ-θ i ) is the Dirac function, which is only valid for θ. i = ψ when the function δ(ψ-θ) i ) = 1, at other times δ(ψ-θ) i ) = 0.

[0110] Equations (13) to (18) are a set of equations describing the motion of the flexible gear. There are a total of 6 equations, which include u, v, w, and φ. x ,φ y ,φ z With 6 variables, it is a fully defined system of partial differential equations that can describe the flexible deformation of a gear under load.

[0111] This embodiment also provides a system dynamics modeling system for meshless flexible gear pairs. This system is used to implement the above embodiments and preferred embodiments, and details already described will not be repeated. As used below, the term "module" can refer to a combination of software and / or hardware that performs a predetermined function. Although the systems described in the following embodiments are preferably implemented in software, hardware implementation, or a combination of software and hardware, is also possible and contemplated.

[0112] This embodiment provides a system dynamics modeling system for meshless flexible gear pairs, such as... Figure 4 As shown, it includes:

[0113] The coordinate establishment module 41 is used to establish the coordinate system of the flexible gear rotor system and calculate the displacement vector of any point of the gear in the coordinate system when the gear rotor system moves.

[0114] The system total kinetic energy calculation module 42 is used to perform time derivative of the displacement vector to obtain the corresponding deformation velocity vector, and calculate the total kinetic energy of the system based on the deformation velocity vector;

[0115] The system's total potential energy calculation module 43 is used to calculate the normal strain potential energy and shear strain potential energy corresponding to the pinion and gear respectively based on the displacement vector, calculate the base boundary condition potential energy corresponding to the pinion and gear respectively based on the displacement vector and base stiffness, and calculate the elastic potential energy formed by gear meshing based on the meshing stiffness and relative position vector of the gear and gear, and sum the various potential energies to obtain the total potential energy of the system.

[0116] The dynamic modeling module 44 substitutes the total kinetic energy and total potential energy of the system into the principles of analytical mechanics, and derives the partial differential control equations describing the system dynamics through variational operations, thus completing the system dynamic modeling of the meshless flexible gear pair.

[0117] Further functional descriptions of the above modules and units are the same as those in the corresponding embodiments described above, and will not be repeated here.

[0118] In this embodiment, the system dynamics modeling system for the meshless flexible gear pair is presented in the form of functional units. Here, a unit refers to an ASIC (Application Specific Integrated Circuit), a processor and memory that execute one or more software or fixed programs, and / or other devices that can provide the above functions.

[0119] This invention also provides a computer device having the above-described features. Figure 4 The system dynamics modeling system of the meshless flexible gear pair is shown.

[0120] Please see Figure 5 , Figure 5 This is a schematic diagram of the structure of a computer device provided in an optional embodiment of the present invention, such as... Figure 5As shown, the computer device includes one or more processors 10, memory 20, and interfaces for connecting the components, including high-speed interfaces and low-speed interfaces. The components communicate with each other via different buses and can be mounted on a common motherboard or otherwise installed as needed. The processors can process instructions executed within the computer device, including instructions stored in or on memory to display graphical information of a GUI on external input / output devices (such as display devices coupled to the interfaces). In some alternative implementations, multiple processors and / or multiple buses can be used with multiple memories and multiple memory modules, if desired. Similarly, multiple computer devices can be connected, each providing some of the necessary operations (e.g., as a server array, a group of blade servers, or a multiprocessor system). Figure 5 Take a processor 10 as an example.

[0121] Processor 10 may be a central processing unit, a network processor, or a combination thereof. Processor 10 may further include a hardware chip. The hardware chip may be an application-specific integrated circuit (ASIC), a programmable logic device (PLD), or a combination thereof. The programmable logic device may be a complex programmable logic device (CAMP), a field-programmable gate array (FPGA), a general-purpose array logic (GDA), or any combination thereof.

[0122] The memory 20 stores instructions executable by at least one processor 10 to cause at least one processor 10 to perform the method shown in the above embodiments.

[0123] The memory 20 may include a program storage area and a data storage area. The program storage area may store the operating system and applications required for at least one function; the data storage area may store data created based on the use of the computer device. Furthermore, the memory 20 may include high-speed random access memory and may also include non-transitory memory, such as at least one disk storage device, flash memory device, or other non-transitory solid-state storage device. In some alternative embodiments, the memory 20 may optionally include memory remotely located relative to the processor 10, and these remote memories may be connected to the computer device via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.

[0124] The memory 20 may include volatile memory, such as random access memory; the memory may also include non-volatile memory, such as flash memory, hard disk or solid-state drive; the memory 20 may also include a combination of the above types of memory.

[0125] The computer device also includes a communication interface 30 for communicating with other devices or communication networks.

[0126] This invention also provides a computer-readable storage medium. The methods described above according to embodiments of the invention can be implemented in hardware or firmware, or implemented as computer code that can be recorded on a storage medium, or implemented as computer code downloaded via a network and originally stored on a remote storage medium or a non-transitory machine-readable storage medium and then stored on a local storage medium. Thus, the methods described herein can be processed by software stored on a storage medium using a general-purpose computer, a dedicated processor, or programmable or dedicated hardware. The storage medium can be a magnetic disk, optical disk, read-only memory, random access memory, flash memory, hard disk, or solid-state drive, etc.; further, the storage medium can also include combinations of the above types of memory. It is understood that computers, processors, microprocessor controllers, or programmable hardware include storage components capable of storing or receiving software or computer code, which, when accessed and executed by the computer, processor, or hardware, implements the methods shown in the above embodiments.

[0127] A portion of this invention can be applied as a computer program product, such as computer program instructions, which, when executed by a computer, can invoke or provide the methods and / or technical solutions according to the invention through the operation of the computer. Those skilled in the art will understand that the forms in which computer program instructions exist in a computer-readable medium include, but are not limited to, source files, executable files, installation package files, etc. Correspondingly, the ways in which computer program instructions are executed by a computer include, but are not limited to: the computer directly executing the instructions, or the computer compiling the instructions and then executing the corresponding compiled program, or the computer reading and executing the instructions, or the computer reading and installing the instructions and then executing the corresponding installed program. Here, the computer-readable medium can be any available computer-readable storage medium or communication medium accessible to a computer.

[0128] Although embodiments of the invention have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of the invention, and such modifications and variations all fall within the scope defined by the appended claims.

Claims

1. A system dynamics modeling method for a meshless flexible gear pair, characterized in that, include: Establish a coordinate system for the flexible gear rotor system, and calculate the displacement vector of any point of the gear in the coordinate system when the gear rotor system moves; The displacement vector is differentiated over time to obtain the corresponding deformation velocity vector, and the total kinetic energy of the gear rotor system is calculated based on the deformation velocity vector. The normal strain potential energy and shear strain potential energy corresponding to the pinion and gear are calculated based on the displacement vector. The base boundary condition potential energy corresponding to the pinion and gear is calculated based on the displacement vector and the base stiffness. The elastic potential energy formed by the meshing of the gear teeth is calculated based on the meshing stiffness and relative position vector of the pinion and gear. The normal strain potential energy, the shear strain potential energy, the base boundary condition potential energy and the elastic potential energy are summed to obtain the total potential energy of the gear rotor system. Substituting the total kinetic energy and total potential energy of the system into the principles of analytical mechanics, the partial differential control equations describing the system dynamics are derived through variational operations, thus completing the system dynamics modeling of the meshless flexible gear pair.

2. The method according to claim 1, characterized in that, The establishment of the coordinate system for the flexible gear rotor system includes: for any point on the flexible gear during system operation, establishing a local coordinate system within the cross-section where the point is located, with the central axis of the cross-section as the origin, and representing the arbitrary point in this local coordinate system as coordinates [x]. i 0z i ], where x i and z i The macroscopic position coordinates of any point in the local coordinate system with the central axis as the origin; Establish a fixed coordinate system, with the center of the flexible gear as the origin of the fixed coordinate system. Transform any point from the local coordinate system to the fixed coordinate system to obtain the coordinates r of the arbitrary point in the fixed coordinate system. 0i =[R i +x i 0z i ], where R i This represents the diameter of the central shaft of the i-th gear.

3. The method according to claim 1 or 2, characterized in that, The displacement vector is obtained by superimposing the coordinates of any point on the gear with the corresponding micro-displacement, wherein the micro-displacement includes micro-lateral displacement in the radial, tangential and axial directions, as well as micro-rotation around the radial, tangential and axial coordinate axes. The expression for the deformation velocity vector includes the gear rotation speed, the partial derivative of the fretting displacement with respect to time, and the partial derivative with respect to the rotation angle.

4. The method according to claim 3, characterized in that, The total kinetic energy of the system is the sum of the kinetic energies of the pinion and gear, respectively. The formula for calculating the kinetic energy of a single gear is: Among them, K i Let R represent the kinetic energy of the i-th gear, where i=1 represents the pinion, i=2 represents the gear, ρ is the density of the gear components, and R is the kinetic energy of the gear. i x represents the diameter of the central shaft of the i-th gear. i This represents the polar coordinate distance from any point on the i-th gear to the central axis. Let A represent the deformation velocity vector of any point on the i-th gear in the coordinate system, A represent the cross-sectional area of ​​the gear, and θ represent the angular integral domain of the gear.

5. The method according to claim 3, characterized in that, The calculation of the normal strain potential energy and shear strain potential energy corresponding to the pinion and gear respectively based on the displacement vector includes: The normal strain and shear strain are derived based on the displacement vector; The normal strain potential energy corresponding to the pinion and gear is calculated based on the following formula: The shear potential energy corresponding to the pinion and gear is calculated based on the following formula: Where i=1 represents the small gear, i=2 represents the large gear, and R i x represents the diameter of the central shaft of the i-th gear. i This represents the polar coordinate distance from any point on the i-th gear to the central axis. Let represent the kinetic energy vector of any point on the i-th gear in a fixed coordinate system, A represent the cross-sectional area of ​​the gear, θ represent the angular integration domain of the gear, and σ represent the kinetic energy vector of any point on the i-th gear. i Let σ represent the normal stress amplitude of the i-th gear component and σ represent the normal stress amplitude of the i-th gear component. i =Eε i , ε i τ is the normal strain of the i-th gear component, and E is the Young's modulus of the material; rt,i ,τ tz,i Let τ represent the shear stress in the radial-tangential plane and the shear stress in the tangential-axial plane of the i-th gear component, and let τ i =μGγ i Where μ is the shear correction factor, G is the shear modulus, and γ rt,i ,γ tz,i Let be the shear strain in the radial-tangential plane and the shear strain in the tangential-axial plane.

6. The method according to claim 5, characterized in that, When calculating normal strain based on displacement vector, the components of the displacement vector at any point on the gear along the radial, tangential, and axial directions are Taylor expanded. After ignoring small quantities of a preset order and above, the expression is obtained. The normal strain is related to the displacement gradient in the radial direction, the displacement gradient in the tangential direction, and the displacement change in the axial direction at that point. When calculating shear strain, only the shear strain in the radial-tangential plane and the shear strain in the tangential-axial plane are considered. The shear strain in the radial-tangential plane is composed of the sum of the partial derivatives of the radial displacement with respect to the tangential coordinate and the partial derivatives of the tangential displacement with respect to the radial coordinate. The shear strain in the tangential-axial plane is composed of the sum of the partial derivatives of the tangential displacement with respect to the axial coordinate and the partial derivatives of the axial displacement with respect to the tangential coordinate.

7. The method according to claim 3, characterized in that, The potential energy of the base boundary conditions for the pinion and the gear is calculated based on the following formula: Where, k ri ,k ti ,k ai Let u be the base stiffness of the i-th gear component. i ,v i ,w i Let represent the radial, tangential, and axial micro-displacements of the i-th gear component, respectively, and D represent the circumferential integral domain of the entire gear, D = [0, 2π). The elastic potential energy generated by gear meshing is calculated based on the following formula: Among them, P m k represents the potential energy of the bearing boundary conditions. m β represents the meshing stiffness between the large gear and the small gear, β represents the helix angle between the large gear and the small gear, v1 is the tangential elastic deformation of the small gear at the meshing position, v2 is the tangential elastic deformation of the large gear at the meshing position, w1 is the axial elastic deformation of the small gear at the meshing position, and w2 is the axial elastic deformation of the large gear at the meshing position.

8. The method according to claim 1, characterized in that, The analytical mechanics principle is Hamilton's principle, which involves substituting the system's total kinetic energy K and total potential energy U into the variational expression of Hamilton's principle. The partial differential equations for each degree of freedom of the meshless gear dynamics were obtained through calculation.

9. A system dynamics modeling system for a meshless flexible gear pair, characterized in that, include: The coordinate establishment module is used to establish the coordinate system of the flexible gear rotor system and calculate the displacement vector of any point of the gear in the coordinate system when the gear rotor system moves. The system total kinetic energy calculation module is used to perform time derivative on the displacement vector to obtain the corresponding deformation velocity vector, and calculate the total kinetic energy of the system based on the deformation velocity vector; The system's total potential energy calculation module is used to calculate the normal strain potential energy and shear strain potential energy corresponding to the pinion and gear respectively based on the displacement vector, calculate the base boundary condition potential energy corresponding to the pinion and gear respectively based on the displacement vector and base stiffness, and calculate the elastic potential energy formed by gear meshing based on the meshing stiffness and relative position vector of the pinion and gear, and sum the various potential energies to obtain the total potential energy of the system. The dynamics modeling module substitutes the total kinetic energy and total potential energy of the system into the principles of analytical mechanics, and derives the partial differential control equations describing the system dynamics through variational operations, thus completing the system dynamics modeling of the meshless flexible gear pair.

10. A computer device, characterized in that, include: The system dynamics modeling method for the meshless flexible gear pair as described in any one of claims 1-8 is provided, with the memory and processor being interconnected and communicating with each other. The memory stores computer instructions, and the processor executes the computer instructions.

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