Rigid-flexible coupling dynamic modeling method, system and storage medium of super-size heavy-duty herringbone gear reduction transmission system

Abstract

The application provides a kind of rigid-flexible coupling dynamics modeling method, system and storage medium of oversized heavy load herringbone gear reduction transmission system, by differentiating rigid-flexible division, oversized driven gear local sector condensation and modular matrix assembly strategy, while accurately characterizing the elastic deformation and coupled dynamics characteristics of large transmission components, effectively control the calculation scale, provide efficient and reliable numerical simulation means for the structure optimization and vibration analysis of oversized herringbone gear transmission system, related to the field of gear transmission system dynamics modeling method, including dividing rigid body and flexible body in the transmission system;Flexible body modeling is carried out on each flexible body;Rigid body modeling is carried out on each rigid body;The connection unit model between each component is established;According to the correspondence relationship of node degrees of freedom, the system overall mass matrix, stiffness matrix and damping matrix are assembled to form a mathematical model for characterizing the dynamics characteristics of transmission system.

Description

Technical Field

[0001] This invention relates to the field of dynamic modeling technology for gear transmission systems, specifically a rigid-flexible coupling dynamic modeling method, system, and storage medium for an ultra-large, heavy-duty herringbone gear reduction transmission system. Background Technology

[0002] Gear transmission systems are core components widely used in industrial equipment, often found in speed reducers. Their dynamic characteristics directly affect the operating performance, reliability, and lifespan of the equipment. With the development of large-scale and heavy-duty equipment in energy, shipbuilding, metallurgy, and mining industries, higher demands are placed on the dynamic performance analysis, vibration response calculation, structural optimization, and fault diagnosis of ultra-large, heavy-duty (torque ≥ 100 kN·m) herringbone gear transmission systems with a center distance ≥ 2000 mm, normal module ≥ 25 mm, and tooth width ≥ 500 mm. Dynamic modeling, as the foundation of the above analysis, directly determines the reliability and engineering practical value of the analysis results through its model accuracy and computational efficiency.

[0003] Currently, existing technologies for dynamic modeling of gear transmission systems can be mainly divided into the following three categories: (I) Pure Rigid Body Modeling Method This method treats all components, including gears, shafts, bearings, and housings, as rigid bodies, neglecting their elastic deformation and considering only rigid body motion (translation and rotation). This method is simple to model and computationally efficient, and is widely used in the preliminary analysis and design of small-to-medium-sized, lightly loaded gear transmission systems. However, for ultra-large, heavy-load gear transmission systems, the flexible deformation of components such as driven gears, output shafts, and housings under heavy loads is significant and cannot be ignored. The pure rigid body assumption leads to large deviations between the results of dynamic performance analysis, vibration response calculation, and modal analysis and the actual situation, making it difficult to meet the requirements of high-precision engineering analysis.

[0004] (II) Fully Flexible Body Modeling Method

[0005] This method uses the finite element method to create flexible body models for all components, preserving a large number of nodal degrees of freedom. Theoretically, a fully flexible body model can achieve the highest computational accuracy and most realistically reflect the structural characteristics of the system. However, for ultra-large gear transmission systems, if all components are modeled as flexible bodies, the system degrees of freedom will reach tens of thousands or even millions, resulting in extremely time-consuming calculations that cannot meet the rapid response requirements of iterative engineering design, parameter optimization, and multi-scheme comparison. Therefore, this method is significantly limited in practical engineering applications.

[0006] (III) Empirical Rule-Based Rigid-Flexible Hybrid Modeling Method

[0007] To strike a balance between computational accuracy and efficiency, existing technologies employ methods that classify some components as flexible bodies while retaining rigid body models for the rest, based on engineering experience. For example, shafts and housings are typically considered flexible bodies, while gears are treated as rigid bodies.

[0008] Chinese patent CN119106573B discloses a gear fault simulation method based on a complex transmission rigid-flexible coupling model. This method uses the Craig-Bampton component modal synthesis method to perform flexible processing and modal condensation on weak stiffness components such as gear rings, gear shafts and housings, and establishes a rigid-flexible coupling model of complex gear transmission systems.

[0009] However, the above-mentioned rigid-flexible hybrid modeling method still has the following shortcomings: First, existing methods do not perform specialized finite element model reduction optimization for ultra-large gears. If a complete finite element model of the driven gear is directly established, its degrees of freedom can reach millions, which is difficult for existing computers to handle. Although CN119106573B uses the Craig-Bampton method to perform modal reduction on the gear ring, its reduction object is the complete gear ring model. The complete model before reduction already contains a massive number of elements of all teeth, resulting in extremely high computational cost and memory consumption during the reduction process. It fails to solve the problem of the excessively large size of the ultra-driven gear model from the source.

[0010] Secondly, existing models often tightly couple the matrices of each component and are constructed "once" without modular matrix assembly, resulting in poor model scalability and difficulty in flexibly adding more complex nonlinear factors such as tooth flank clearance, time-varying stiffness, and lubrication models.

[0011] In summary, existing technologies, when applied to the dynamic modeling of ultra-large, heavy-duty herringbone gear transmission systems, generally suffer from a lack of dedicated finite element method for reducing the order of the super-driven gear and poor model scalability. Therefore, there is an urgent need to propose a dynamic modeling method that can control the size of the super-driven gear model from the source and employs a modular matrix assembly approach to meet the high-precision engineering analysis requirements of ultra-large, heavy-duty herringbone gear reduction transmission systems. Summary of the Invention

[0012] The technical problem to be solved by this invention is to provide a rigid-flexible coupling dynamic modeling method, system and storage medium for ultra-large size heavy-duty herringbone gear reduction transmission system. By differentiating rigid-flexible partitioning, local sector condensation of ultra-large driven gears and modular matrix assembly strategy, the method can effectively control the computational scale while accurately characterizing the elastic deformation and coupling dynamic characteristics of large transmission components, and provide an efficient and reliable numerical simulation means for the structural optimization and vibration analysis of ultra-large size herringbone gear transmission system.

[0013] The technical solution adopted by this invention to solve its technical problem is: a rigid-flexible coupling dynamic modeling method for ultra-large size heavy-duty herringbone gear reduction transmission system. This method is applicable to ultra-large size herringbone gear transmission systems with a center distance ≥ 2000 mm, normal module ≥ 25 mm, and tooth width ≥ 500 mm, and includes the following steps: Obtain the geometric and material parameters of the input shaft, output shaft, driven gear, driving gear, and gearbox in the transmission system, and obtain the boundary conditions of the transmission system. The boundary conditions include at least the bearing installation position and bearing support stiffness. According to the preset rigid-flexibility division criteria, the rigid and flexible bodies in the transmission system are divided; wherein, the rigid-flexibility division criteria are: the output shaft, gearbox housing and driven gear are divided into flexible bodies, and the input shaft and driving gear are divided into rigid bodies. Based on the geometric and material parameters of each of the defined flexible bodies, a flexible body model is created for each flexible body. The finite element method is used to establish its mass matrix and stiffness matrix, and a model reduction method is used to reduce the model order, thereby obtaining the condensed mass matrix, stiffness matrix, and master node information of each flexible body. Among them, the flexible body model of the driven gear adopts a local sector model with N teeth, where N is an integer between 3 and 7. The model reduction process is performed with the hole center and tooth meshing position of the driven gear as the master nodes to reduce the model size of the driven gear. Based on the geometric and material parameters of each rigid body, a rigid body model is performed for each rigid body, and the degree of freedom information of each node of each rigid body is obtained. Based on the obtained master node information of each flexible body after condensation and the obtained degree of freedom information of each node of each rigid body, a connection unit model between each component is established, and the matrix of each connection unit is obtained. The connection unit model includes at least a gear pair meshing unit, a bearing support unit, and a shaft-gear connection unit. The stiffness parameters of the connection unit model are determined according to the geometric parameters and material parameters of the corresponding component or a preset stiffness value. The bearing support unit is established according to the bearing support stiffness. Based on the obtained mass and stiffness matrices of each flexible body after condensation, the mass and stiffness matrices of each rigid body, and the matrix of each connecting unit, the system's overall mass matrix, stiffness matrix, and damping matrix are assembled according to the correspondence of nodal degrees of freedom, forming a mathematical model for characterizing the dynamic characteristics of the transmission system.

[0014] Furthermore, the geometric parameters of the input shaft and output shaft include at least the diameter and length of each shaft segment; the geometric parameters of the driven gear and driving gear include at least the number of teeth, normal module, tooth width, helix angle, and pressure angle; the geometric parameters of the gearbox include at least the center position of the bearing hole and the external dimensions of the gearbox. The material parameters include density and elastic modulus.

[0015] Furthermore, the flexible body modeling of the output shaft is performed by reducing the model order using the bearing center, driven gear center, and output end of the output shaft as the main nodes; The flexible body modeling of the gearbox adopts a hybrid modeling of shell elements and solid elements, and the model is reduced in order by using the bearing hole center and the center of both sides of the gearbox as the main nodes.

[0016] Furthermore, N=5, each herringbone tooth is composed of two tooth surfaces with opposite helix angles along the axial direction, and three main nodes are set along the tooth width direction at the intersection of the center line of each tooth surface and the pitch circle, for a total of 30 main nodes, and one main node is set at the center of the driven gear hole; each of the above main nodes has 6 degrees of freedom.

[0017] Furthermore, the flexible body modeling of the output shaft adopts a hybrid modeling method using beam elements and solid elements.

[0018] Furthermore, the flexible body modeling of the gearbox adopts a hybrid modeling method using shell elements and solid elements.

[0019] Furthermore, the input shaft's mass matrix and stiffness matrix are established using the spatial beam element method, and the drive gear's mass matrix and stiffness matrix are established using the lumped mass method.

[0020] Furthermore, the gear pair meshing unit includes a time-varying meshing stiffness matrix and a meshing damping matrix; wherein, the time-varying meshing stiffness matrix includes torsional meshing stiffness and axial coupling stiffness, and the axial coupling stiffness is (tanβ)² times the torsional meshing stiffness, where β is the helix angle of the herringbone gear.

[0021] Furthermore, the bearing support unit uses a linear spring-damping unit to simulate the supporting effect of each rolling bearing, and the bearing support stiffness includes radial support stiffness and axial support stiffness. The shaft-gear connection unit uses the penalty function method to simulate the rigid connection between the shaft and the gear.

[0022] Furthermore, the model reduction method is either Guyan's condensation method or Craig-Bampton's component modal synthesis method.

[0023] A rigid-flexible coupling dynamic modeling system for ultra-large heavy-duty herringbone gear reduction transmission system is used to implement the rigid-flexible coupling dynamic modeling method for the ultra-large heavy-duty herringbone gear reduction transmission system mentioned above. It includes: a parameter acquisition module, a rigid-flexible partitioning module, a flexible body modeling module, a rigid body modeling module, a connection unit modeling module, and a matrix assembly module.

[0024] A computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the above-described rigid-flexible coupling dynamic modeling method for an ultra-large-size heavy-duty herringbone gear reduction transmission system.

[0025] The beneficial effects of this invention are as follows: This invention provides a rigid-flexible coupled dynamic modeling scheme that balances computational accuracy and solution efficiency for ultra-large-sized, heavy-duty herringbone gear reduction transmission systems with a center distance ≥2000mm, normal module ≥25mm, and tooth width ≥500mm. This scheme first incorporates the significantly deformable output shaft, gearbox, and driven gear into flexible bodies through differentiated rigid-flexible partitioning criteria, while treating the input shaft and driving gear as rigid bodies. This ensures that key elastic deformations are fully accounted for while effectively controlling the total number of system degrees of freedom, avoiding the enormous computational burden associated with fully flexible modeling. Secondly, for the massive driven gears, this method uses a local sector model containing only 3 to 7 teeth for finite element flexible body modeling, and only uses the hole center and tooth meshing position as the master node for model reduction. While fully preserving the tooth surface contact deformation coordination information and constraint boundary information, it significantly reduces the matrix size of the gear substructure, avoiding the dual computational burden of building a complete model and condensing it. By significantly reducing the matrix size of the gear substructure while fully preserving the tooth surface contact deformation coordination information and constraint boundary information, the fine dynamic simulation of ultra-large gears achieves engineering-acceptable computational efficiency, demonstrating significant technical advantages in finite element reduction of ultra-large gears. Furthermore, this scheme abandons the traditional tightly coupled modeling approach. It employs a multi-type modular connection matrix construction strategy, establishing gear pair meshing units, bearing support units, and shaft-gear connection units. Each connection unit is independently determined based on the actual geometric material parameters of the component or a preset stiffness, and then the overall matrix is ​​assembled according to the nodal degrees of freedom correspondence. This significantly improves the model's scalability and flexibility, allowing complex nonlinear factors such as tooth backlash, time-varying stiffness, and lubrication effects to be easily embedded into the model. This effectively solves the technical shortcomings of existing models, which are difficult to modify and expand due to integrated construction. Finally, all the reduced-order flexible body matrices, rigid body matrices, and connection unit matrices are precisely assembled into the overall system mass matrix, stiffness matrix, and damping matrix through degree-of-freedom correspondence, forming a complete dynamic mathematical model that can be directly used for natural frequency analysis, harmonic response analysis, transient response analysis, and stability determination. This provides a high-fidelity, solvable, and easily scalable numerical simulation foundation for the structural optimization, vibration suppression, fault diagnosis, and reliability assessment of ultra-large-size heavy-duty herringbone gear reduction transmission systems. Attached Figure Description

[0026] Figure 1 This is a structural schematic diagram of an ultra-large, heavy-duty herringbone gear reduction transmission system; Figure 2This is a flowchart of the present invention; Figure 3 This is a finite element model of a driven gear with 5 teeth. The diagram shows: gearbox 1, input shaft 2, driving gear 3, output shaft 4, driven gear 5, and bearing 6. Detailed Implementation

[0027] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0028] like Figure 1 As shown, the ultra-large heavy-duty herringbone gear reduction transmission system includes a gearbox 1, an input shaft 2, a driving gear 3, an output shaft 4, a driven gear 5, and bearings 6 for mounting the input shaft and the output shaft.

[0029] like Figure 2 As shown, the present invention discloses a rigid-flexible coupling dynamic modeling method for ultra-large-size heavy-duty herringbone gear reduction transmission systems. This method is applicable to ultra-large-size herringbone gear transmission systems with a center distance ≥ 2000 mm, a normal module ≥ 25 mm, and a tooth width ≥ 500 mm. The method includes the following steps: S1. Obtain the geometric and material parameters of the input shaft, output shaft, driven gear, driving gear, and gearbox in the gear transmission system, and obtain the boundary conditions of the gear transmission system. The boundary conditions include at least the bearing installation position and bearing support stiffness.

[0030] Specifically, bearing support stiffness includes radial stiffness and axial stiffness. Radial stiffness and axial stiffness are determined according to bearing model, rolling element parameters and preload conditions. Their sources include one or a combination of the following methods: (1) Calculation using the theoretical formula of rolling bearing support stiffness - based on Hertz contact theory, the bearing stiffness is obtained by differentiating the load-deformation relationship according to the diameter, number, contact angle and load conditions of the rolling elements; (2) Determination from bearing manufacturer's sample parameters - the stiffness parameters of the corresponding bearing model are obtained from the bearing manufacturer's product catalog or technical manual; (3) Experimental determination - the axial and radial displacements of the bearing under different loads are measured on the rolling bearing loading test bench, and the bearing stiffness is obtained from the load-displacement curve.

[0031] Geometric and material parameters are used for subsequent modeling. Specifically, in this invention, the geometric parameters of the input and output shafts include at least the diameter and length of each shaft segment; the geometric parameters of the driven and driving gears include at least the number of teeth, normal module, tooth width, helix angle, and pressure angle; the geometric parameters of the gearbox include at least the center position of the bearing holes and the external dimensions of the gearbox; and the material parameters include density and elastic modulus.

[0032] S2. According to the preset rigid-flexibility division criteria, the rigid and flexible bodies in the transmission system are divided; specifically, the rigid-flexibility division criteria are: the output shaft, gearbox housing and driven gear are divided into flexible bodies, and the input shaft and driving gear are divided into rigid bodies. S3. Based on the corresponding geometric and material parameters of each of the divided flexible bodies, a flexible body model is performed for each flexible body. The finite element method is used to establish its mass matrix and stiffness matrix, and a model reduction method is used to reduce the model order to obtain the condensed mass matrix, stiffness matrix and master node information of each flexible body. Among them, the flexible body model of the driven gear adopts a local sector model with N teeth, where N is an integer between 3 and 7. The model reduction process is performed with the hole center and tooth meshing position of the driven gear as the master nodes to reduce the model size of the driven gear.

[0033] In some embodiments, the flexible body modeling of the output shaft can employ a pure beam element model, a pure solid element model, a SIMBEAM modal beam model, or a lumped mass-spring model; the flexible body modeling of the gearbox can employ a pure shell element model, a pure solid element model, or a substructure / super-element method. To control the overall model size while ensuring the computational accuracy of key components, and to avoid the explosion of degrees of freedom in a pure solid model or insufficient accuracy in a pure beam / shell model, preferably in this invention, the flexible body modeling of the output shaft uses a hybrid modeling of beam elements and solid elements; the flexible body modeling of the gearbox uses a hybrid modeling of shell elements and solid elements.

[0034] Specifically, the main body of the output shaft uses beam elements to ensure computational efficiency, and the key parts connecting the output shaft and the driven gear use solid elements to ensure stress calculation accuracy; the thin-walled main body of the gearbox uses shell elements to ensure computational efficiency, and the locally thick areas such as the bearing seats and mounting bosses of the gearbox use solid elements to ensure local stiffness and stress calculation accuracy.

[0035] To reduce the condensed degrees of freedom of the driven gear to a controllable range while ensuring the accuracy of the herringbone gear meshing characteristics, and to avoid the explosion of degrees of freedom in the complete finite element model leading to computational infeasibility, see [reference needed]. Figure 3Preferably, in this invention, N=5, each herringbone tooth includes two tooth surfaces with opposite helix angles along the axial direction, and each tooth surface has 3 main nodes at the intersection of the pitch circle and the tooth surface centerline, on both sides and in the middle along the tooth width direction; therefore, 5 herringbone teeth form a total of 5×2×3=30 tooth surface main nodes, and 1 main node is set at the center of the driven gear hole; each of the above main nodes has 6 degrees of freedom. Comparative experiments have verified that the calculation error of the 3-tooth local model is 10.2%, while the calculation error of the 5-tooth local model can be controlled within 2.1%, and the degrees of freedom of the driven gear after condensation is only 186, which can effectively control the model size while ensuring calculation accuracy.

[0036] Preferably, the output shaft has five main nodes: the bearing center (2 nodes), the driven gear center (1 node), and the shaft output end (2 nodes). Each main node has six degrees of freedom, and after condensation, a 30×30 mass matrix and stiffness matrix are obtained. The gearbox has five main nodes: the input and output shaft bearing hole centers (4 nodes) and the centers on both sides of the gearbox (1 node). Each main node has six degrees of freedom, and after condensation, a 30×30 mass matrix and stiffness matrix are obtained.

[0037] Those skilled in the art should understand that the number and position of the aforementioned master nodes are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. The selection of the aforementioned master nodes follows the general principle of "determining master nodes based on the position of the mechanical interaction interface" in the Guyan condensation method—the bearing center corresponds to the point of application of the bearing support force, the driven gear center corresponds to the point of transmission of the gear meshing force, and the output end corresponds to the point of application of the load torque, ensuring that the condensed model can accurately transmit support constraints, meshing torque, and load torque, while effectively controlling the model size. Furthermore, the condensation method is not limited to the Guyan method; in other embodiments, Craig-Bampton component modal synthesis, IRS improved condensation system method, or super-element method can also be used. As long as the flexible body can be condensed to retain the master nodes, these are all alternative embodiments of the present invention.

[0038] S4. Based on the geometric and material parameters of each rigid body, perform rigid body modeling for each rigid body and obtain the degree of freedom information of each node of each rigid body.

[0039] In some embodiments, the input shaft can be modeled using a pure rigid body model, and the driving gear can be modeled using the lumped mass method. The input shaft is a slender rotating body structure. Although it is classified as a rigid body, to accurately describe its spatial mass inertia distribution, it is preferable to use the spatial beam element method to establish its equivalent mass matrix. This method can accurately characterize the translational and rotational inertia characteristics of a rigid body in space using one-dimensional elements, significantly reducing the degrees of freedom compared to solid element modeling. The driving gear is a disc-shaped structure, and the influence of its internal elastic deformation on the system's dynamic characteristics is negligible. Using the lumped mass method to establish its equivalent mass matrix requires only 6 degrees of freedom to fully describe its translational and rotational inertia characteristics, making modeling the simplest and most efficient method.

[0040] Furthermore, to facilitate the numerical assembly and solution of the overall system matrix, the input shaft and the driving gear do not contribute physical elastic stiffness to the overall stiffness matrix. Instead, their rigid body properties are characterized by introducing numerical penalty stiffness or rigid constraint equations to ensure sufficient numerical stability within the system matrix. Therefore, during the assembly of the overall system matrix, the mass matrices of the input shaft and the driving gear (generated by the beam element method and the lumped mass method respectively) participate in the superposition of the system mass matrix, while their equivalent stiffness matrix adopts the large stiffness method (stiffness value set to 10 of the surrounding flexible elements). 6 The model is filled with a factor of 1 (x) to approximate rigid constraints at the numerical level, thereby reducing modeling complexity and computational cost while ensuring computational accuracy.

[0041] Specifically, the input shaft is divided into multiple spatial beam elements, each with 6 degrees of freedom. The mass matrix and stiffness matrix of each beam element are calculated based on the shaft segment diameter and length, and then assembled into the overall matrix of the input shaft. The driving gear adopts a lumped mass model with 6 degrees of freedom, and its mass matrix is ​​a diagonal matrix. ; Where m is the mass of the driving gear. , , These represent the moments of inertia of the driving gear about the three coordinate axes.

[0042] By combining the two methods, the total degrees of freedom for rigid body modeling is only 12. This minimizes the contribution of rigid body to the total degrees of freedom of the system while ensuring computational accuracy, allowing computational resources to be concentrated on the detailed modeling of flexible body, thus balancing computational accuracy and efficiency.

[0043] S5. Based on the obtained master node information of each flexible body after condensation and the obtained degree of freedom information of each node of each rigid body, establish a connection unit model between each component and obtain the matrix of each connection unit. The connection unit model includes at least a gear pair meshing unit, a bearing support unit and a shaft-gear connection unit. The stiffness parameters of the connection unit model are determined according to the geometric parameters and material parameters of the corresponding component or a preset stiffness value. The bearing support unit is established according to the bearing support stiffness.

[0044] Specifically, the gear pair meshing unit includes a time-varying meshing stiffness matrix and a meshing damping matrix; for the herringbone gear structure, the time-varying meshing stiffness matrix includes torsional meshing stiffness and axial coupling stiffness. The torsional meshing stiffness is calculated using the energy method, simplifying the gear teeth as variable cross-section cantilever beams on the tooth root circle. Bending stiffness, shear stiffness, axial compressive stiffness, Hertzian contact stiffness, and gear ring base stiffness are calculated separately. These stiffness components are connected in series to obtain the meshing stiffness of a single pair of teeth. Then, based on the overlap ratio, the stiffnesses of multiple pairs of teeth meshing simultaneously are superimposed in parallel, and the average value is taken over one meshing cycle. In this embodiment, the average torsional meshing stiffness is 6.4 × 10⁻⁶. 9 N / m. The axial coupling stiffness is derived based on the geometric relationship of the helix angle of the herringbone gear. The ratio of axial stiffness to torsional stiffness is (tanβ)², where β is the helix angle of the herringbone gear (β=22° in this embodiment), that is, the axial coupling stiffness is (tanβ)² times the torsional stiffness.

[0045] The bearing support unit and shaft-gear connection unit are key bridges that couple the condensed independent substructures into a complete system dynamic model based on their physical connections. They are used to construct a complete load transfer path from gear meshing excitation through the driven gear, output shaft, bearing, and housing, ensuring that the system stiffness matrix is ​​non-singular and the dynamic equations are solvable. To achieve effective coupling of the condensed substructures while maintaining the physical accuracy of the connection relationships, preferably in this invention, the bearing support unit includes radial and axial support stiffness matrices. Each rolling bearing is simplified as a linear spring-damping unit, with its radial and axial stiffness as preset stiffness values ​​input, established based on the bearing support stiffness obtained in step S1. In this embodiment, the radial stiffness is taken as 2.5 × 10⁻⁶. 9 N / m, added to the x and y degrees of freedom of the corresponding bearing node; axial stiffness is taken as 1.2 × 10⁻⁶. 9 N / m, added to the z-direction degree of freedom of the corresponding bearing node.

[0046] The shaft-gear connection unit uses the penalty function method to simulate a rigid connection. The penalty stiffness, as a preset stiffness value, is taken to be much larger than the maximum stiffness component of the system (at least an order of magnitude higher) to ensure that the connection is approximately rigid. The purpose of setting the penalty stiffness is to introduce a rigid constraint relationship between the independently established shaft substructure and gear substructure, so that they satisfy the displacement compatibility condition at the connection node, avoiding the singularity of the system stiffness matrix due to lack of constraints, thereby ensuring that the dynamic equations are solvable. At the same time, this approach allows the shaft and gear to be modeled as independent substructures, coupled only through the penalty function during the matrix assembly stage. This avoids the model adjustment difficulties caused by having to model the shaft and gear as a whole to meet the connection constraints, thus supporting the overall modular assembly architecture of this invention. In this embodiment, the penalty stiffness is taken as 1×10. 9 N / m is added to the stiffness matrix of the corresponding degrees of freedom between the input shaft and the driving gear, and between the output shaft and the driven gear, respectively, to simulate the rigid constraints of shaft-gear interference fit or spline connection.

[0047] In this invention, the preset stiffness values ​​include, but are not limited to, the bearing support stiffness and penalty stiffness mentioned above. Those skilled in the art can select the corresponding preset stiffness value input according to the actual connection unit type.

[0048] S6. Based on the obtained mass and stiffness matrices of each of the flexible bodies after condensation, the mass and stiffness matrices of each of the rigid bodies, and the matrices of each of the connecting units, they are assembled into the overall system mass matrix, stiffness matrix, and damping matrix according to the correspondence of the nodal degrees of freedom, forming a mathematical model for characterizing the dynamic characteristics of the transmission system.

[0049] Before assembly, a unified global degree-of-freedom (DOF) numbering system is first established. In this embodiment, the total DOF of the system is 258, of which: driven gear (after condensation) 186 DOF, output shaft (after condensation) 30 DOF, gearbox (after condensation) 30 DOF, input shaft (rigid body) 6 DOF, and driving gear (rigid body) 6 DOF. Each master node has 6 degrees of freedom in three-dimensional space, namely three translational degrees of freedom x, y, z and three rotational degrees of freedom θx, θy, θz.

[0050] After the global degrees of freedom are uniformly numbered according to the order of "flexible bodies first, then rigid bodies," the sub-matrices are superimposed onto the corresponding rows and columns of the overall system matrix according to the nodal degrees of freedom correspondence. Specifically, the condensed mass and stiffness matrices of the driven gear, output shaft, and gearbox, the mass and stiffness matrices of the input shaft and driving gear, and the connection matrices of the gear pair meshing unit, bearing support unit, and shaft-gear connection unit are respectively superimposed onto the corresponding degrees of freedom positions of the overall system mass and stiffness matrices. The radial and axial stiffness contributions of the bearing support unit are added to the corresponding bearing node degrees of freedom; the penalty function stiffness contribution of the shaft-gear connection unit is added to the corresponding degree of freedom at the shaft-gear connection. The overall system damping matrix is ​​calculated using the Rayleigh damping formula. C = α · M + β · K ; Where α and β are Rayleigh damping coefficients, determined based on the first two natural frequencies of the system and the corresponding modal damping ratios.

[0051] The overall system mass matrix obtained after assembly M Stiffness matrix K and damping matrix C Together they constitute the following system dynamics mathematical model: ; Where u is the system's generalized displacement vector, which contains displacement variables of all retained degrees of freedom of the system; Let be the system's generalized velocity vector, which is the first derivative of the displacement vector with respect to time. F(t) is the generalized acceleration vector of the system, which is the second derivative of the displacement vector with respect to time; F(t) is the external excitation vector, including time-varying loads such as gear meshing force and input torque.

[0052] The above system overall quality matrix M and stiffness matrix K The results are obtained by superimposing the submatrices according to their degrees of freedom: M = M1+M2+M3+M4+M5+M6 ; K = K1+K2+K3+K4+K5+K6+K7+K8 ; in, M1 , M2 , M3 The mass matrices are the condensed mass matrices of the three flexible bodies: driven gear, output shaft and gearbox housing, with dimensions of 186×186, 30×30 and 30×30, respectively. M4 , M5The mass matrices for the two rigid bodies, the input shaft and the drive gear, are 6×6 respectively. M6 The contribution to the quality matrix of each connection unit is negligible in this embodiment. K1 , K2 , K3 These are the stiffness matrices of the three flexible bodies—driven gear, output shaft, and gearbox—after condensation. K4 , K5 These are the stiffness matrices of the two rigid bodies, the input shaft and the driving gear, respectively. K6 , K7 , K8 These are the stiffness matrix contributions for the gear pair meshing unit, bearing support unit, and shaft-gear connection unit, respectively. Each sub-matrix is ​​superimposed onto the corresponding row and column of the overall system matrix according to its corresponding position in the global degree of freedom number. During assembly, the dimensions of each matrix are expanded to 258×258 according to the global degree of freedom number.

[0053] This invention also provides a rigid-flexible coupling dynamic modeling system for an ultra-large heavy-duty herringbone gear reduction transmission system, used to implement the above-mentioned rigid-flexible coupling dynamic modeling method for the ultra-large heavy-duty herringbone gear reduction transmission system, characterized in that it includes: The parameter acquisition module is used to execute step S1; The rigid-flexible partitioning module is used to execute step S2; The flexible body modeling module is used to execute step S3; Rigid body modeling module, used to execute step S4; The connection unit modeling module is used to perform step S5; and The matrix assembly module is used to perform step S6.

[0054] The present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the above-described rigid-flexible coupling dynamic modeling method for an ultra-large-size heavy-duty herringbone gear reduction transmission system.

[0055] Example: For ultra-large heavy-duty herringbone gear reduction transmission systems, eight typical parameter combinations covering the range of center distance 2000-4000mm, normal module 25-50mm, tooth width 500-1100mm, and input torque 100000-500000N·m were selected for comparative verification. The specific parameters of each example are shown in Table 1.

[0056] Table 1 Summary of Instance Parameters

[0057] To verify the advantages of this invention in controlling model size, the following three modeling schemes were compared: Scheme A (the method of this invention): The driven gear uses a 5-tooth local sector model with Craig-Bampton condensation, the output shaft and housing use Guyan condensation, and the driving gear and input shaft are modeled as rigid bodies. Scheme B (fully flexible body modeling): All components use complete finite element models without order reduction. Scheme C (existing rigid-flexible coupling method): The driven gear uses a complete finite element model with Craig-Bampton condensation, and the rest is the same as Scheme A.

[0058] The comparison of the degree of freedom scale of each scheme under 8 experimental cases is shown in Table 2.

[0059] Table 2 Comparison of Degrees of Freedom for Different Modeling Schemes

[0060] Table 3 shows a comparison of the computation time of each instance under the same computing environment (Intel Xeon Gold 6254 3.1GHz processor, 128GB memory, 8-core parallel processing).

[0061] Table 3 Comparison of computation time for different modeling schemes

[0062] As can be seen from Tables 2 and 3, the method of the present invention significantly reduces the computational scale and significantly improves computational efficiency.

[0063] To verify the model accuracy of the method of the present invention, the calculation results of the fully flexible body modeling (Scheme B) were used as the reference benchmark (considered as the "true value"), and the calculation errors of Scheme A and Scheme C were compared. The vibration response amplitudes at the first three natural frequencies and meshing frequencies of each experimental example were selected as accuracy evaluation indicators, and the comparison results are shown in Tables 4 and 5.

[0064] Table 4 Comparison of calculation errors for the first three natural frequencies of each example (relative to the fully flexible model)

[0065] Table 5 Comparison of vibration response amplitude error at meshing frequency for each example (relative to the fully flexible model)

[0066] As shown in Tables 4 and 5, the average errors of the first three natural frequencies of the method of the present invention (Scheme A) are 1.36%, 1.96%, and 2.28%, respectively, all controlled within 3%; the average error of the vibration response amplitude is 2.68%, controlled within 5%, meeting the engineering accuracy requirements. The existing rigid-flexible coupling method (Scheme C) has an average error of 5.03%–8.25% in natural frequencies and an average error of 9.50% in vibration response amplitude, both significantly higher than the method of the present invention.

[0067] To verify the effectiveness of the method of the present invention in modal analysis, the ratio of the first three natural frequencies to the meshing frequency of each example was calculated, and the results are shown in Table 6.

[0068] Table 6. Relationship between modal frequencies and meshing frequencies for each experimental example

[0069] As shown in Table 6, the first three natural frequencies of all examples maintain a safety margin of more than 20% with the meshing frequency, with no risk of resonance, which verifies the engineering practical value of the method of the present invention in modal analysis.

Claims

1. A rigid-flexible coupling dynamic modeling method for ultra-large size heavy-duty herringbone gear reduction transmission systems, the method being applicable to ultra-large size herringbone gear transmission systems with a center distance ≥2000mm, normal module ≥25mm, and tooth width ≥500mm, characterized in that... Includes the following steps: Obtain the geometric and material parameters of the input shaft, output shaft, driven gear, driving gear, and gearbox in the transmission system, and obtain the boundary conditions of the transmission system. The boundary conditions include at least the bearing installation position and bearing support stiffness. According to the preset rigid-flexibility division criteria, the rigid and flexible bodies in the transmission system are divided; wherein, the rigid-flexibility division criteria are: the output shaft, gearbox housing and driven gear are divided into flexible bodies, and the input shaft and driving gear are divided into rigid bodies. Based on the geometric and material parameters of each of the defined flexible bodies, a flexible body model is created for each flexible body. The finite element method is used to establish its mass matrix and stiffness matrix, and a model reduction method is used to reduce the model order, thereby obtaining the condensed mass matrix, stiffness matrix, and master node information of each flexible body. Among them, the flexible body model of the driven gear adopts a local sector model with N teeth, where N is an integer between 3 and 7. The model reduction process is performed with the hole center and tooth meshing position of the driven gear as the master nodes to reduce the model size of the driven gear. Based on the geometric and material parameters of each rigid body, a rigid body model is performed for each rigid body, and the degree of freedom information of each node of each rigid body is obtained. Based on the obtained master node information of each flexible body after condensation and the obtained degree of freedom information of each node of each rigid body, a connection unit model between each component is established, and the matrix of each connection unit is obtained. The connection unit model includes at least a gear pair meshing unit, a bearing support unit, and a shaft-gear connection unit. The stiffness parameters of the connection unit model are determined according to the geometric parameters and material parameters of the corresponding component or a preset stiffness value. The bearing support unit is established according to the bearing support stiffness. Based on the obtained mass and stiffness matrices of each flexible body after condensation, the mass and stiffness matrices of each rigid body, and the matrix of each connecting unit, the system's overall mass matrix, stiffness matrix, and damping matrix are assembled according to the correspondence of nodal degrees of freedom, forming a mathematical model for characterizing the dynamic characteristics of the transmission system.

2. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The geometric parameters of the input shaft and output shaft include at least the diameter and length of each shaft segment; the geometric parameters of the driven gear and driving gear include at least the number of teeth, normal module, tooth width, helix angle, and pressure angle; the geometric parameters of the gearbox include at least the center position of the bearing hole and the external dimensions of the gearbox. The material parameters include density and elastic modulus.

3. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The flexible body modeling of the output shaft uses the bearing center, driven gear center, and output end of the output shaft as the main nodes to perform model reduction processing. The flexible body modeling of the gearbox adopts a hybrid modeling of shell elements and solid elements, and the model is reduced in order by using the bearing hole center and the center of both sides of the gearbox as the main nodes.

4. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The N=5, each herringbone tooth is composed of two tooth surfaces with opposite helix angles along the axial direction, and three main nodes are set along the tooth width direction at the intersection of the center line of each tooth surface and the pitch circle, for a total of 30 main nodes. One main node is set at the center of the driven gear hole; each of the above main nodes has 6 degrees of freedom.

5. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The flexible body modeling of the output shaft adopts a hybrid modeling method using beam elements and solid elements.

6. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The flexible body modeling of the gearbox uses a hybrid modeling approach combining shell elements and solid elements.

7. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The mass matrix and stiffness matrix of the input shaft are established using the spatial beam element method, and the mass matrix and stiffness matrix of the driving gear are established using the lumped mass method.

8. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The gear pair meshing unit includes a time-varying meshing stiffness matrix and a meshing damping matrix; wherein, the time-varying meshing stiffness matrix includes torsional meshing stiffness and axial coupling stiffness, and the axial coupling stiffness is (tanβ)² times the torsional meshing stiffness, where β is the helix angle of the herringbone gear.

9. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The bearing support unit uses a linear spring-damping unit to simulate the support function of each rolling bearing, and the bearing support stiffness includes radial support stiffness and axial support stiffness. The shaft-gear connection unit uses the penalty function method to simulate the rigid connection between the shaft and the gear.

10. The rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in claim 1, characterized in that, The model reduction method is either Guyan's condensation method or Craig-Bampton's component modal synthesis method.

11. A rigid-flexible coupling dynamic modeling system for an ultra-large heavy-duty herringbone gear reduction transmission system, used to implement the rigid-flexible coupling dynamic modeling method for an ultra-large heavy-duty herringbone gear reduction transmission system as described in any one of claims 1 to 10, characterized in that, include: The parameter acquisition module is used to acquire the geometric and material parameters of the input shaft, output shaft, driven gear, driving gear and gearbox in the transmission system, and to acquire the boundary conditions of the transmission system. The boundary conditions include at least the installation position of the bearing and the bearing support stiffness. The rigid-flexible partitioning module is used to partition the rigid and flexible bodies in the transmission system according to a preset rigid-flexible partitioning criterion; wherein, the rigid-flexible partitioning criterion is: the output shaft, gearbox housing and driven gear are partitioned as flexible bodies, and the input shaft and driving gear are partitioned as rigid bodies. The flexible body modeling module is used to model each flexible body based on its geometric and material parameters. It employs the finite element method to establish the mass and stiffness matrices, and then uses a model reduction method to reduce the model's order, obtaining the condensed mass and stiffness matrices and their master node information for each flexible body. Specifically, the flexible body modeling of the driven gear uses a local sector model with N teeth, where N is an integer between 3 and 7. The model reduction process uses the hole center and tooth meshing position of the driven gear as master nodes to reduce the model size of the driven gear. The rigid body modeling module is used to perform rigid body modeling on each of the rigid bodies based on the geometric and material parameters of each rigid body, and to obtain the degree of freedom information of each node of each rigid body. A connection unit modeling module is used to establish connection unit models between components based on the obtained master node information of each flexible body after condensation and the obtained degree-of-freedom information of each node of each rigid body, and to obtain the matrix of each connection unit. The connection unit model includes at least gear meshing units, bearing support units, and shaft-gear connection units. The stiffness parameters of the connection unit models are determined according to the geometric and material parameters of the corresponding components or preset stiffness values. The bearing support units are established based on the bearing support stiffness. The matrix assembly module is used to assemble the overall system mass matrix, stiffness matrix, and damping matrix according to the corresponding nodal degrees of freedom, based on the obtained mass and stiffness matrices of each of the flexible bodies, the mass and stiffness matrices of each of the rigid bodies, and the matrices of each of the connecting units, to form a mathematical model for characterizing the dynamic characteristics of the transmission system.

12. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the rigid-flexible coupling dynamic modeling method for the ultra-large size heavy-duty herringbone gear reduction transmission system as described in any one of claims 1 to 10.

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