CN110807278B - Three-dimensional solid unit modeling method of gear system - Google Patents
Three-dimensional solid unit modeling method of gear system Download PDFInfo
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Abstract
According to the structural characteristics of a gear transmission system, HYPERMESH software is utilized to conduct pretreatment of a complex structure, a three-dimensional solid unit model of a gear substructure and a transmission shaft substructure is respectively built, and precise division and optimization of grids are conducted; then, carrying out modal analysis by utilizing ANSYS software and extracting a mass and rigidity matrix; next, modeling gear tooth meshing of the gear transmission system by utilizing a Pasternak elastic foundation beam theory in consideration of nonlinear factors such as time-varying rigidity and the like; finally, according to interface displacement coordination conditions, integrating, coupling and dimension reduction are carried out on the models of the transmission shaft, the gear and the bearing system by using a mode synthesis method, and a high-precision gear transmission system dynamics model which is matched with the dynamic characteristics of equipment is established; the method can ensure the model precision and remarkably improve the calculation efficiency, thereby providing a more efficient and accurate method for the quantitative analysis of the vibration of the gear transmission system.
Description
Technical Field
The invention belongs to the technical field of dynamic analysis of gear systems, and particularly relates to a three-dimensional solid unit modeling method of a gear system.
Background
The gear transmission has the characteristics of compact structure, stable transmission ratio, high transmission efficiency and the like, and is widely applied to various industrial mechanical equipment to play an irreplaceable role. As a key component for transmitting power and energy, the vibration characteristics of the gear system directly affect the working accuracy, noise and working reliability of the whole mechanical system. Therefore, the dynamic analysis through the gear system is a key for understanding whether the dynamic characteristics of the system are reasonably designed. The gear transmission system is a complex dynamic system, and establishing a dynamic model of the gear transmission system is a central task for carrying out gear dynamic analysis. Only if a correct and reasonable dynamics model is established, the dynamics of the gear transmission system can be effectively analyzed. At present, the method for establishing the dynamic model of the gear transmission system mainly comprises the following steps: a concentrated parameter method, a transfer matrix method, a finite element method and the like. The centralized parameter method is to make the gear system equivalent by adopting a simple spring mass system, and is generally only suitable for qualitative analysis. The advantage of the hierarchical matrix method is that the order of the matrix does not increase with the increase of the degree of freedom of the system, and the disadvantage is that the coupling analysis between the subsystems is inconvenient when the higher order frequency is calculated or the rotor system is large. The three-dimensional finite element method does not carry out a great deal of simplification treatment on the structure, has accurate model and is suitable for quantitative analysis. However, the dynamic model equation established by adopting the three-dimensional finite element method has high order and is difficult to solve, and is generally only suitable for statics or quasi-statics analysis.
Disclosure of Invention
In order to overcome the problems of the prior art, the invention aims to provide a three-dimensional entity unit modeling method of a gear system, which is based on a three-dimensional finite element modeling theory, a Pasternak elastic foundation beam theory and a modal synthesis theory, can rapidly and accurately describe vibration response characteristics of a gear transmission system on the premise of ensuring accuracy, reveals a vibration response mechanism of the gear system, provides theoretical support for quantitative analysis of the gear system, can ensure model accuracy, can obviously improve calculation efficiency, and provides a more efficient and accurate analysis method for quantitative analysis of vibration of the gear transmission system.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
a method of modeling a three-dimensional solid unit of a gear system, comprising the steps of:
1) Respectively establishing three-dimensional models of all the substructures of the driving shaft in the gear system through three-dimensional modeling software SOLIWORKS; grid division and optimization are carried out by utilizing HYPERMESH software; performing modal analysis by using ANSYS software and extracting a mass and rigidity matrix; according to the interface displacement coordination condition, a fixed interface mode synthesis method is used for carrying out degree of freedom reduction and model integration on the finite element model;
2) Modeling the engagement of the driving wheels and the driven wheels through the Pasternake foundation Liang Moxing and Hertz contact theory, and simulating the engagement relationship between the driving wheels and the driven wheels; deriving a rigidity matrix of the foundation beam model according to a Lagrangian equation; and integrally coupling the meshing contact model and the master-slave power model to obtain a dynamic model of the gear transmission system.
The specific steps of the step 1) comprise:
(1) According to the structural characteristics of the gear transmission shaft, the driving shaft is divided into two sub-structures: the gear substructure and the transmission shaft substructure are subjected to grid division and optimization by utilizing HYPERMESH software, modal analysis is performed by utilizing ANSYS software, and mass M under physical coordinates of each transmission shaft substructure is extracted s Stiffness matrix K s And mass M in physical coordinates of the gear substructure g Stiffness matrix K g ；
(2) Dividing physical coordinates of a transmission shaft substructure into internal node coordinates and interface node coordinates, taking a shaft hole of a gear as a connecting interface of the two substructures, taking the interface coordinates as interface node coordinates, fixing all the interface coordinates, and obtaining a branch characteristic problem equation of a fixed interface:
wherein: phi s The main mode of the transmission shaft subsystem is reserved,
-stiffness matrix corresponding to the internal nodes of the drive shaft, < >>
-mass moment, ω, -natural frequency of the system, corresponding to the external node of the drive shaft;
let the modal matrix phi s The method comprises the following steps:
wherein: psi s -constrained primary mode of drive shaft subsystem, i. -identity matrix;
performing physical transformation by using a modal matrix shown in formula (2), and transforming physical coordinates of the substructure into modal coordinates:
wherein:
-physical coordinates of nodes inside the substructure, +.>
-physical coordinates of the interface nodes of the substructure, +.>
-master modal coordinate, +.>
-constraining the modality coordinates of the modality;
converting the mass, stiffness matrix and force vector in the physical coordinate system into the mass in the modal coordinate system by using the modal matrix
Rigidity matrix->
Force vector->
Wherein: f (f) s -force vector under physical coordinate system. Phi sT -modality matrix Φ s Is a transposed matrix of (a);
the mass matrix, the rigidity matrix and the force vector of the gear substructure under the modal coordinates can be obtained by the same method
(3) Rewriting mode coordinates of a transmission shaft and a gear substructure into
Wherein (1)>
-node coordinates reserved for connecting the bearing units, < - > j->
-node coordinates reserved for the wheels for coupling the driving and driven wheels, "jm" -contact interface coordinates of the gear and the shaft;
according to the geometric coordination condition of the connection of the two substructures, the non-independent modal coordinates are converted into independent modal coordinates, and the mass matrix M of the driving shaft system (comprising the transmission shaft and the gear) under the independent modal coordinates z Stiffness matrix K z External force vector f z The mass matrix M of the driven shaft system (comprising the transmission shaft and the gear) can be obtained by the same method c Stiffness matrix K c External force vector f c 。
The specific steps of the step 2) comprise:
(1) Modeling gear tooth rigidity: assuming that n contact lines exist on the tooth surface, each contact line can be simulated by a foundation beam, the stiffness coefficient of the foundation beam changes with time and the change of the engagement position, and the strain energy of the ith foundation quantity model can be expressed as:
by using the equation, the rigidity matrix K of the foundation beam model can be deduced according to the Lagrange equation dd ；
(2) Modeling gear tooth engagement: it was simulated using discrete spring units, assuming that the contact between the teeth of the wheels is a hertz contact, the stiffness of the discrete spring units can be expressed as:
wherein Deltaeta is the distance between equivalent spring units; E. v-Young's modulus and Poisson's ratio;
(3) The integration of the system model, the displacement relation of the nodes between the master gear system and the slave gear system is utilized, the assembly is carried out according to the idea of finite elements, and the dynamics equation of the assembled model can be expressed as follows:
wherein: d, displacement vector of elastic foundation; m is M qq 、K qq Q-mass matrix, stiffness matrix and displacement vector of the three-dimensional entity unit of the gear transmission shaft system; k (K) dd -a stiffness matrix of the elastic foundation; k (K) dq -coupling stiffness matrix of three-dimensional solid model and elastic foundation model, K dq -coupling stiffness matrix of three-dimensional solid model and elastic foundation model, K qd -matrix K dq Transposed matrix f of (f) d Force vector due to gear transmission error, f q -an input-output torque acting on the input system;
transforming the formula (7) in a static polycondensation mode, wherein a system dynamics equation after static polycondensation can be rewritten as follows:
wherein:
the rigidity matrix of the system after static polycondensation,
-external force load of the system after static polycondensation; damping-Rayleigh damping C t,q ＝βK t,q Beta-damping coefficient.
Compared with the prior art, the invention has the beneficial effects that:
the three-dimensional entity unit dynamics modeling method of the gear transmission system provided by the invention comprehensively utilizes the three-dimensional finite element modeling theory, the Pasternak elastic foundation beam theory and the modal synthesis theory to establish an accurate dynamics model of the gear transmission system. The method can ensure the model precision, can obviously improve the calculation efficiency, and can effectively solve the contradiction between the solving efficiency and the model precision in the dynamic modeling of the gear system. The method provided by the invention can provide a more efficient and accurate analysis method for quantitative analysis of vibration of the gear transmission system.
Drawings
Fig. 1 is a finite element model of a gear drive shaft and a division of substructures.
Fig. 2 is a gear tooth foundation Liang Moxing.
Fig. 3 is a tooth engagement model.
Fig. 4 is a three-dimensional kinetic model of the gear train.
FIG. 5 is a finite element model of a gear drive shaft system, and FIG. 5 (a) is a finite element model of a drive wheel; 5 (b) is the overall tooth width load; fig. 5 (c) is the intermediate node bearing load.
Fig. 6 is a node displacement curve, and fig. 6 (a) is a load applied to all nodes (f=100deg.N); fig. 6 (b) is a load applied to the middle 3 nodes (f=70n).
Fig. 7 is a time domain simulation waveform of a gear train.
Detailed Description
The invention is further described below with reference to the drawings and examples.
A method of modeling a three-dimensional solid unit of a gear system, comprising the steps of:
1) Respectively establishing three-dimensional models of all the substructures through three-dimensional modeling software SOLIWORKS; grid division and optimization are carried out by utilizing HYPERMESH software; performing modal analysis by using ANSYS software and extracting a mass and rigidity matrix; and (3) radically researching interface displacement coordination conditions, and carrying out degree of freedom reduction and model integration on the finite element model by using a fixed interface mode synthesis method.
2) Modeling the engagement of the driving wheels and the driven wheels through the Pasternake foundation Liang Moxing and Hertz contact theory, and simulating the engagement relationship between the driving wheels and the driven wheels; deriving a rigidity matrix of the foundation beam model according to a Lagrangian equation; and integrally coupling the meshing contact model and the master-slave power model to obtain a dynamic model of the gear transmission system.
The specific steps of the step 1) comprise:
(1) As shown in fig. 1, the driving shaft is divided into two sub-structures according to the structural characteristics of the gear transmission shaft itself: the gear substructure and the transmission shaft substructure are subjected to grid division and optimization by utilizing HYPERMESH software, modal analysis is performed by utilizing ANSYS software, and mass M under physical coordinates of each transmission shaft substructure is extracted s Stiffness matrix K s And mass M in physical coordinates of the gear substructure g Stiffness matrix K g 。
(2) The physical coordinates of the transmission shaft substructure are divided into internal node coordinates and interface node coordinates, the shaft hole of the gear is used as a connecting interface of the two substructures, and the interface coordinates are used as interface node coordinates. Dividing the physical coordinates of the internal nodes of the transmission shaft and the physical coordinates of the interface nodes into a module matrix form, and obtaining a undamped motion equation of the substructure of the transmission shaft, wherein the undamped motion equation comprises:
wherein:
-physical coordinates of internal nodes of the transmission shaft, +.>
-physical coordinates of the transmission shaft interface nodes, +.>
-interface force->
-stiffness matrix corresponding to the internal nodes of the drive shaft, < >>
-mass matrix corresponding to external nodes of the transmission shaft, < >>
-a coupling quality matrix corresponding to the internal node and the interface node,>
-a coupling quality matrix corresponding to the internal node and the interface node.
Fixing all interface coordinates to obtain a branch characteristic problem equation of a fixed interface:
wherein: phi s The main mode of the transmission shaft subsystem is reserved, and omega is the natural frequency of the system.
Let the modal matrix phi s The method comprises the following steps:
wherein: psi s -constrained primary mode of drive shaft subsystem, I-identity matrix.
Using the matrix to perform physical transformation, transforming the physical coordinates of the substructure to modal coordinates:
wherein: :
-physical coordinates of nodes inside the substructure, +.>
-physical coordinates of the interface nodes of the substructure, +.>
-master modal coordinate, +.>
The modality coordinates of the modality are constrained.
Converting the mass, stiffness matrix and force vector in the physical coordinate system into the mass in the modal coordinate system by using the modal matrix
Rigidity matrix->
Force vector->
Wherein: f (f) s -force vector under physical coordinate system.
The mass matrix, the rigidity matrix and the force vector of the gear substructure under the modal coordinates can be obtained by the same method
(3) Rewriting mode coordinates of a transmission shaft and a gear substructure into
Wherein (1)>
-node coordinates reserved for connecting the bearing units, < - > j->
The coordinates of the nodes reserved by the wheel bodies for coupling the driving wheel and the driven wheel, and the coordinates of the contact interface between the gear and the shaft.
According to the geometric coordination condition of the connection of the two substructures, the non-independent modal coordinates are converted into independent modal coordinates, and the mass matrix, the rigidity matrix and the external force vector of the driving shaft system (comprising the transmission shaft and the gear) under the independent modal coordinates are as follows:
s in the formula is a modal coordinate transformation matrix.
The specific steps of the step 2) comprise:
(1) Modeling gear tooth rigidity: as shown in fig. 2, for gear engagement, a foundation beam model may be used to simulate the structural deformation of the gear. Assuming that n contact lines exist on the tooth surface, each contact line can be simulated by a foundation beam, and the rigidity coefficient of the foundation beam changes with time and the change of the engagement position. The strain energy of the ith foundation quantity model can be expressed as:
using the above equation, it can be deduced from the Lagrangian equationRigidity matrix K of foundation beam model dd 。
(2) Modeling gear tooth engagement: after the stiffness matrix of the elastic foundation beam model for simulating the structural deformation of the driving gear and the driven gear is obtained by using the formula (0-1), the foundation beam models of the driving gear and the driven gear are required to be connected so as to simulate the meshing of gear teeth. As shown in fig. 3, a discrete spring unit was used to simulate it. Assuming that the contact between the teeth of the wheels is a hertz contact, the stiffness of the discrete spring units can be expressed as:
wherein Deltaeta is the distance between equivalent spring units; E. v-Young's modulus and Poisson's ratio;
(3) Integration of system models: a three-dimensional solid element finite element model of the gear train is shown in fig. 4. The model consists of three parts: (1) a gear drive shafting model; (2) foundations Liang Moxing; (3) a centralized parametric model to simulate bearing support. The displacement relation of the nodes between the main driven gear system and the driven gear system is utilized, the assembly is carried out according to the idea of finite elements, and the assembled model dynamics equation can be expressed as follows:
wherein: d, displacement vector of elastic foundation; m is M qq 、K qq Q-mass matrix, stiffness matrix and displacement vector of the three-dimensional entity unit of the gear transmission shaft system; k (K) dd -a stiffness matrix of the elastic foundation; k (K) dq -coupling stiffness matrix of three-dimensional solid model and elastic foundation model, K qd -matrix K dq Transposed matrix f of (f) d Force vector due to gear transmission error, f q -input-output torque acting on the input system.
The mass matrix of formula (18) is a singular matrix, which can be transformed by means of static polycondensation. The system dynamics equation after static polycondensation can be rewritten as:
wherein:
-stiffness matrix of the system after static polycondensation, < ->
-external force load of the system after static polycondensation; damping-Rayleigh damping C t,q ＝βK t,q Beta-damping coefficient.
Description of the preferred embodiments
To verify the validity of the built model, solid models of the gear drive system drive shaft system as shown in the figure are built by SOLIWORKS and HYPERMESH, and finite element cell meshing is performed. And then, a degree-of-freedom reduction model of the model is built by using the method, tens of thousands of degrees of freedom are arranged before the degree of freedom is reduced, and hundreds of degrees of freedom are arranged after the degree of freedom is reduced, so that the solving speed of a dynamics equation is improved. The model is subjected to comparative analysis from three angles of mode, static response and dynamic response.
Modal analysis: a finite element model of the drive shaft system of the gear train is shown in fig. 5 (a). Firstly, applying fixed constraint at a node where a bearing is located, and carrying out modal analysis on the model by utilizing ANSYS to obtain the natural frequency of the model. And then, carrying out feature solving on the model with the degree of freedom reduced by using the mode synthesis method to obtain the natural frequency of the system (when using the mode synthesis method, the first 50 steps are taken to reserve the main mode). The first 10 th order natural frequencies of the system obtained using these two methods are shown in table 1. From this table, it can be seen that: the natural frequencies of the systems obtained by the two methods are very close. The method has the advantages that the equation order is greatly reduced by using the C-B fixed interface mode synthesis method, the calculation efficiency is improved, and meanwhile, accurate low-order mode information is reserved.
Table 1 natural frequency contrast
Static response: as shown in fig. 5 (b) and (c), forces are applied to all the nodes of the tooth width of the gear and the middle three nodes, and the applied loads are 100N and 70N respectively. The node displacement curve obtained by the statics analysis module of ANSYS software and the node displacement curve obtained by the elastic foundation beam model are shown in fig. 6. From the graph, the node displacement curves obtained by the two methods are quite close; under the action of two load applying modes, the maximum node displacement error of the results obtained by the two methods is 3.93% and 4.72% respectively.
Dynamic response: in the dynamic response analysis, the gear input shaft is assumed to have a frequency of f=8 Hz, and the gear train has a mesh frequency fp=440 Hz. And solving a system dynamics equation by using a Newmark-beta method, and obtaining a vibration displacement response curve of the gear transmission system at a bearing center node as shown in figure 7. As can be seen, the vibration response of the normal gear system is composed of two alternating periodic impulse responses of unequal strength, the period of each two strong impulse or each two weak impulse vibration responses is 0.00227s, and the frequency is 440Hz which is equal to the meshing frequency of the gear system. The meshing principle of gears shows that the impact vibration response is mainly caused by single-double-tooth alternate meshing of gear teeth of a gear transmission system, and the simulation result is consistent with the theoretical analysis result.
Claims (1)
1. A method for modeling a three-dimensional solid unit of a gear system, comprising the steps of:
1) Respectively establishing three-dimensional models of all the substructures of the driving shaft in the gear system through three-dimensional modeling software SOLIWORKS; grid division and optimization are carried out by utilizing HYPERMESH software; performing modal analysis by using ANSYS software and extracting a mass and rigidity matrix; according to the interface displacement coordination condition, a fixed interface mode synthesis method is used for carrying out degree of freedom reduction and model integration on the finite element model;
2) Modeling the engagement of the driving wheels and the driven wheels through the Pasternake foundation Liang Moxing and Hertz contact theory, and simulating the engagement relationship between the driving wheels and the driven wheels; deriving a rigidity matrix of the foundation beam model according to a Lagrangian equation; the meshing contact model and the master-slave power model are integrated and coupled to obtain a dynamic model of the gear transmission system;
the specific steps of the step 1) comprise:
(1) According to the structural characteristics of the gear transmission shaft, the driving shaft is divided into two sub-structures: the gear substructure and the transmission shaft substructure are subjected to grid division and optimization by utilizing HYPERMESH software, modal analysis is performed by utilizing ANSYS software, and mass M under physical coordinates of each transmission shaft substructure is extracted s Stiffness matrix K s And mass M in physical coordinates of the gear substructure g Stiffness matrix K g ；
(2) Dividing physical coordinates of a transmission shaft substructure into internal node coordinates and interface node coordinates, taking a shaft hole of a gear as a connecting interface of the two substructures, taking the interface coordinates as interface node coordinates, fixing all the interface coordinates, and obtaining a branch characteristic problem equation of a fixed interface:
wherein: phi s The main mode of the transmission shaft subsystem is reserved,
-stiffness matrix corresponding to the internal nodes of the drive shaft, < >>
-mass moment, ω, -natural frequency of the system, corresponding to the external node of the drive shaft;
let the modal matrix phi s The method comprises the following steps:
wherein: psi s -constrained primary mode of drive shaft subsystem, i. -identity matrix;
performing physical transformation by using a modal matrix shown in formula (2), and transforming physical coordinates of the substructure into modal coordinates:
wherein:
-physical coordinates of nodes inside the substructure, +.>
-physical coordinates of the interface nodes of the substructure, +.>
-master modal coordinate, +.>
-constraining the modality coordinates of the modality;
converting the mass, stiffness matrix and force vector in the physical coordinate system into the mass in the modal coordinate system by using the modal matrix
Rigidity matrix->
Force vector->
Wherein: f (f) s -force vector under physical coordinate system; phi sT -modality matrix Φ s Is a transposed matrix of (a);
the mass matrix, the rigidity matrix and the force vector of the gear substructure under the modal coordinates can be obtained by the same method
/>
(3) Rewriting mode coordinates of a transmission shaft and a gear substructure into
Wherein (1)>
-node coordinates reserved for connecting the bearing units, < - > j->
-node coordinates reserved for the wheels for coupling the driving and driven wheels, "jm" -contact interface coordinates of the gear and the shaft;
according to the geometric coordination condition of the connection of the two substructures, the non-independent modal coordinates are converted into independent modal coordinates, and the mass matrix M of the driving shaft system under the independent modal coordinates z Stiffness matrix K z External force vector f z The mass matrix M of the driven shaft system can be obtained by the same method c Stiffness matrix K c External force vector f c ；
The specific steps of the step 2) comprise:
(1) Modeling gear tooth rigidity: assuming that n contact lines exist on the tooth surface, each contact line can be simulated by a foundation beam, the stiffness coefficient of the foundation beam changes with time and the change of the engagement position, and the strain energy of the ith foundation quantity model can be expressed as:
by using the equation, the rigidity matrix K of the foundation beam model can be deduced according to the Lagrange equation dd ；
(2) Modeling gear tooth engagement: it was simulated using discrete spring units, assuming that the contact between the teeth of the wheels is a hertz contact, the stiffness of the discrete spring units can be expressed as:
wherein Deltaeta is the distance between equivalent spring units; E. v-Young's modulus and Poisson's ratio;
(3) The integration of the system model, the displacement relation of the nodes between the master gear system and the slave gear system is utilized, the assembly is carried out according to the idea of finite elements, and the dynamics equation of the assembled model can be expressed as follows:
wherein: d, displacement vector of elastic foundation; m is M qq 、K qq Q-mass matrix, stiffness matrix and displacement vector of the three-dimensional entity unit of the gear transmission shaft system; k (K) dd -a stiffness matrix of the elastic foundation; k (K) dq -coupling stiffness matrix of three-dimensional solid model and elastic foundation model, K dq -coupling stiffness matrix of three-dimensional solid model and elastic foundation model, K qd -matrix K dq Transposed matrix f of (f) d Force vector due to gear transmission error, f q -an input-output torque acting on the input system;
transforming the formula (7) in a static polycondensation mode, wherein a system dynamics equation after static polycondensation can be rewritten as follows:
wherein:
-stiffness matrix of the system after static polycondensation, < ->
-external force load of the system after static polycondensation; damping-Rayleigh damping C t,q ＝βK t,q Beta-damping coefficient. />
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| CN103995940A (en) * | 2014-06-05 | 2014-08-20 | 清华大学 | Drive axle dynamic characteristic computing method considering input torque changing |
| CN106960093A (en) * | 2017-03-22 | 2017-07-18 | 清华大学 | A kind of transmission system method for numerical simulation for considering gear and bearing Non-linear coupling |
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