Dynamic modeling method of flexible bevel gear system considering meshing point change

Abstract

The invention belongs to the technical field of gear transmission, and discloses a dynamic modeling method of a flexible bevel gear system considering meshing point change, which is used for analyzing vibration response and dynamic stress response characteristic laws of the bevel gear system in a traveling wave resonance state. And further analyzing the traveling wave resonance phenomenon of the spiral bevel gear system in the aero-engine accessory casing. A three-dimensional shell unit and a beam unit are adopted to establish a bevel gear pair finite element entity model. A dynamic model of the flexible bevel gear system considering meshing point changes is established by adopting a substructure modal synthesis method and a rotation modal projection method. The reliability of the provided model is verified through a finite element entity model and an experimental test result. On the basis, dynamic response and vibration stress experiment tests are combined, and the traveling wave resonance phenomenon of the bevel gear system is researched. The research result of the provided model has theoretical significance on design and fault monitoring of the aero-engine bevel gear system.

Description

Technical Field

[0001] The invention relates to the technical field of gear transmission, and in particular to a dynamic modeling method of a flexible bevel gear system taking into account the change of meshing points. Background Art

[0002] The spiral bevel gear system is one of the key components in the accessory casing of an aircraft engine. The working performance of the aircraft engine system is very sensitive to the operating accuracy of the bevel gear system. In the design process of the aircraft engine gear system, it is often necessary to consider the requirements of lightweight, so the accessory casing bevel gear system is often designed as a thin rim / thin web. This design form often leads to traveling wave resonance caused by pitch diameter vibration during the operation of the bevel gear system, and matrix crack failure occurs, which seriously affects the operation of the aircraft engine system. Existing research on the traveling wave resonance of aircraft engine gear systems is mostly focused on the analysis of the pitch diameter vibration of the gear system through experiments. The analysis content includes the pitch diameter modal frequency, vibration mode and dynamic stress analysis of the gear under test. In addition, the analysis of the traveling wave resonance phenomenon of the gear system through simulation calculation is mostly focused on the analysis of the pitch diameter mode of the gear system and the axial vibration of the gear matrix. The analysis of the vibration response characteristics of the gear system in the traveling wave resonance state needs to be further deepened. At the same time, most simulation studies on the traveling wave resonance of gear systems use the finite element method to establish a gear wheel model and analyze its axial vibration response law. Comprehensively considering, the solution accuracy and solution efficiency of the dynamic model established by this method have a large room for improvement. Therefore, it is necessary to establish a new dynamic model to solve the above technical problems. Summary of the invention

[0003] The purpose of the present invention is to address the deficiencies in the prior art and to propose a dynamic modeling method for a flexible bevel gear system that takes into account the change in meshing points, thereby improving the calculation efficiency while ensuring accuracy.

[0004] To achieve the above object, the technical solution of the present invention is as follows: A dynamic modeling method of a flexible bevel gear system considering the change of meshing point comprises the following steps:

[0005] Step 1: Based on the parameters of the spiral bevel gear, the structural parameters of the spiral bevel gear, the structural parameters of the driving wheel, and the structural parameters of the driven wheel, a three-dimensional 8-node shell element is used to simulate the bevel gear base, and a Timoshenko beam element is used to simulate the rotating shaft. A rigid connection between the inner hole of the bevel gear base and the rotating shaft is established through an interface coupling unit to obtain a finite element solid model of the bevel gear pair;

[0006] Step 2: Based on the finite element solid model of the bevel gear pair, the time-varying meshing stiffness of the bevel gear pair is solved under the state of root circle constraint, and the nodes on the base columns of the two bevel gear bases are connected by meshing springs;

[0007] Step 3: Consider the stress of the spiral bevel gear system during the meshing process, analyze the stress on the driving wheel and the driven wheel of the spiral bevel gear, and obtain the dynamic meshing force;

[0008] Step 4: According to the time-varying meshing stiffness obtained in step 2 and the dynamic meshing force obtained in step 3, the dynamic model of the flexible spiral bevel gear system is obtained;

[0009] Step 5: Based on the dynamic model of the flexible spiral bevel gear system obtained in step 4, the substructure modal synthesis method is introduced, and the physical coordinates are transformed into modal coordinates through the transformation matrix to obtain the vibration differential equation of the condensed flexible bevel gear system model;

[0010] Step 6: In order to simulate the actual meshing state between gear pairs more realistically, based on the vibration differential equation of step 5, the changes of the meshing teeth involved in the meshing process are considered, and a dynamic model of the flexible arc bevel gear system considering the changes of the meshing point during the actual rotation of the gear is established. When the driving wheel and the driven wheel are condensed, the number of main nodes of the gear matrix retained is the corresponding number of teeth. When the driving wheel and the driven wheel rotate and pass through each tooth node in turn, the rotational mode projection and coordinate transformation equations are introduced to simulate the influence of the gear rotation on the coupling between the two elements, and the dynamic model of the gear system considering the changes of the meshing point during the actual operation is obtained.

[0011] Furthermore, the step 1 is specifically as follows:

[0012] The flexible bevel gear matrix is ​​modeled by finite element using MATLAB software, and a unit mass matrix is ​​established; the unit mass matrix is ​​expressed as:

[0013]

[0014] Where N is the shape parameter matrix, ρ is the gear density, and J is the Jacobian matrix;

[0015] The element stiffness matrix is ​​expressed as:

[0016]

[0017] Where B is the strain matrix and D is the constitutive matrix;

[0018] Unit gyro matrix G b e It is expressed as:

[0019]

[0020] In the formula, ψ, θ t and φ are the rotation angles along the x, y, and z directions, respectively;

[0021] The Timoshenko beam element is used to characterize the rotating shaft model, and its element mass matrix M s e It is expressed as:

[0022]

[0023] In the formula, A S is the cross-sectional area of ​​each shaft unit, J is the polar area moment of inertia of each shaft unit, ρ is the material density, l is the length of each shaft unit; a, b, c, d, f and g are variables;

[0024] Element stiffness matrix K s e It is expressed as:

[0025]

[0026] Where G is the shear modulus, E is the elastic modulus, and h, i, j, and k are as follows:

[0027]

[0028] Where I is the diameter moment of inertia, φ is the shear influence factor;

[0029] Unit gyro matrix G s e It is expressed as:

[0030]

[0031] Where Ω is the angular velocity of rotation, and p, q, s, and w are as follows:

[0032]

[0033] In the formula, Furthermore, the step 2 is specifically as follows:

[0034] O s1 and O s2 is the connection node on the base column of the two bevel gears; considering that the 8-node shell element contains translational freedom and rotational freedom, on the basis of constraining the gear base circle, ignoring the rotational freedom generated during meshing, the O s1 and O s2 The meshing spring between the two meshing planes does not project the torsional stiffness or translational stiffness excitation onto the meshing plane. The unit stiffness matrix is ​​calculated using the projection method. On this basis, the generalized displacement vector is expressed as:

[0035] q m =[x p1 ,y p1 ,z p1 ,x g2,y g2 ,z g2 ]

[0036] x p1 Represents the x-direction freedom of the driving wheel; g2 represents the x-direction freedom of the driven wheel; y p1 Represents the y-direction freedom of the driving wheel; g2 represents the y-direction freedom of the driven wheel; z p1 Represents the axial freedom of the driving wheel; z g2 Represents the axial freedom of the driven wheel;

[0037] The corresponding projection vector is expressed as:

[0038]

[0039] In the formula, α b is the pressure angle, β b is the helix angle, δ p and δ g are the pitch cone angle of the driving wheel and the pitch cone angle of the driven wheel respectively;

[0040] Therefore, the stiffness matrix of the bevel gear pair meshing unit under the constrained gear base circle state is obtained:

[0041]

[0042] Where k(t) is the time-varying mesh stiffness of the spiral bevel gear pair;

[0043] In the process of stiffness matrix assembly, the stiffness matrix of the bevel gear pair meshing unit is expressed as:

[0044]

[0045] Furthermore, considering the stress of the spiral bevel gear system during the meshing process, the force analysis of the driving wheel and the driven wheel of the spiral bevel gear is performed. In step 3, the normal meshing force of the gear is decomposed into component forces along each coordinate axis, specifically:

[0046] During the operation of the gear system, vibration will cause strong nonlinear effects. The dynamic model of the flexible bevel gear system is usually only applicable to the case of no meshing loss. In the present invention, the gear torque is kept below 150N·m during the operation of the bevel gear system. Therefore, there is no meshing loss during the operation. Therefore, the present invention does not consider the influence of nonlinear meshing backlash. n Projected onto the x, y, and z axes respectively, we get the component force F of the driving wheel x1 、F y1 、F z1 and the component force F of the driven wheel x2 、Fy2 、F z2 , respectively expressed as:

[0047]

[0048] Furthermore, in step 4, the dynamic model of the flexible spiral bevel gear system is expressed as:

[0049]

[0050] In the formula, ζ f is the overall displacement vector, F is the external load vector, M, C, G, and K are the overall mass matrix, overall damping matrix, overall gyro matrix, and overall stiffness matrix of the flexible spiral bevel gear system, respectively, which are expressed as:

[0051] M=diag(M f_p ,M f_g )

[0052] G=diag(G f_p ,G f_g )

[0053]

[0054] Where M f_p 、M f_g are the overall mass matrices of the driving wheel and the driven wheel respectively; G f_p , G f_g are the overall gyro matrices of the driving wheel and the driven wheel, respectively, K f_p , K f_g is the overall stiffness matrix of the driving wheel and the driven wheel, expressed as;

[0055]

[0056] In the formula, K p and K g are the stiffness matrices of the driving wheel and the driven wheel respectively; K σ_p and K σ_g are the centrifugal stiffness matrices of the driving wheel and the driven wheel, K S_p and K S_g are the rotation softening matrices of the driving wheel and the driven wheel respectively; C is the overall damping matrix;

[0057] The stiffness matrix of the support bearing is expressed as:

[0058] K b =diag(k bx ,k by ,k bz ,k bθx ,k bθy ,kbθz )

[0059] In the formula, k bx , k by ,,k bz ,,k bθx ,,k bθy and k bθz Respectively represent the translation stiffness in the x, y, z directions and the rotation stiffness around the x, y, z axes;

[0060] The support stiffness matrices of the driving wheel and the driven wheel are expressed as:

[0061]

[0062] The overall mass matrix M is composed of the overall mass matrix of the driving wheel and the driven wheel; the overall stiffness matrix K is composed of the stiffness matrix of the gear matrix and the shaft unit of the driving wheel and the driven wheel, the bevel gear pair meshing unit, the bearing unit and the coupling unit; the stiffness matrix of the bevel gear matrix and the shaft is composed of the stiffness matrix of the three-dimensional 8-node shell unit and the Timoshenko beam unit respectively. The gear matrix model and the shaft model are coupled by constraining the inner hole nodes of the bevel gear. The coupling unit is established between the driving / driven gear matrix and the shaft through the rigid connection between the nodes;

[0063] Two supporting bearings are respectively arranged on the driving wheel and the driven wheel rotating shaft.

[0064] Furthermore, compared with the finite element model, the bevel gear model of the gear base and the shaft is simulated by shell elements and beam elements, which greatly improves the solution efficiency. On this basis, in order to further improve the calculation efficiency, the substructure modal synthesis (CMS) method is introduced to transform the physical coordinates into modal coordinates through the transformation matrix. The specific step 5 is:

[0065] A=M or K

[0066] Where, M and K are the overall mass matrix and stiffness matrix of the bevel gear system, respectively, S is the transformation matrix; the superscript “-” indicates the condensation matrix;

[0067] S is composed of the main mode matrix Φ f and the constraint mode matrix Φ b Composition, expressed as:

[0068] S=[Φ f Φ b ],

[0069] The main modal matrix refers to the modal eigenvector when the main node is constrained; the constrained modal matrix refers to the displacement vector obtained by applying a unit displacement to the degree of freedom of a main node and constraining the degrees of freedom of all other main nodes, expressed as:

[0070]

[0071] Where I is the unit matrix, V is the displacement vector from the node, and F is the external force;

[0072] Select the bevel gear base nodes and shaft segment nodes of the driving wheel and the driven wheel as the main nodes of the bevel gear system respectively;

[0073] The vibration differential equation of the flexible bevel gear system model after polycondensation is expressed as:

[0074]

[0075] In the formula, They are the overall mass matrix, overall damping matrix, overall gyro matrix and overall stiffness matrix of the flexible spiral bevel gear system after polycondensation. is the external load vector after polycondensation;

[0076] The stiffness matrix of the dynamic model after polycondensation is expressed as:

[0077]

[0078] In the formula, the subscripts "p" and "g" are the driving wheel and the driven wheel respectively, M f_p and M f_g They are the overall stiffness matrices of the flexible driving wheel and driven wheel of the bevel gear model respectively;

[0079] The overall rotation matrix of the kinetic model after polycondensation is expressed as:

[0080]

[0081] In the formula, G f_p and G f_g are the overall rotation matrices of the bevel gear flexible driving wheel and driven wheel respectively;

[0082] After the driving wheel and the driven wheel are condensed, the stiffness matrix of the bevel gear pair meshing unit is rewritten as:

[0083]

[0084] After the driving wheel and the driven wheel are condensed, the stiffness matrix of the supporting bearing is rewritten as:

[0085]

[0086] The overall stiffness matrix of the dynamic model after polycondensation is expressed as:

[0087]

[0088] In the formula, K f_p and K f_g They are the overall stiffness matrices of the flexible driving wheel and driven wheel of the bevel gear model respectively;

[0089] The damping matrix of the kinetic model after polycondensation is expressed as:

[0090]

[0091] In the formula, C α and C β are the Rayleigh damping coefficients, respectively;

[0092] The meshing external forces of the driving wheel and the driven wheel of the kinetic model after polycondensation are expressed as:

[0093] f e =[f ep f eg ] T ,

[0094] f ep 、f eg They are the meshing external forces of the driving wheel and the driven wheel of the flexible spiral bevel gear system, is the meshing external force of the driving wheel and the driven wheel after condensation;

[0095] The external load of the kinetic model after polycondensation is expressed as:

[0096]

[0097] Furthermore, as the gears run, the meshing gear teeth are also changing. Existing studies usually equate the meshing between gear pairs to a form of constant meshing points connected by springs. In order to simulate the actual meshing state between gear pairs more realistically, the changes of the meshing gear teeth involved in the meshing process are considered, and a dynamic model of a flexible arc bevel gear system is established that considers the changes in the meshing points during the actual rotation of the gears. In step 6, the main node numbers of the gear matrix are retained when the driving wheel and the driven wheel are condensed to the corresponding number of teeth. When the driving wheel and the driven wheel rotate and pass through each tooth node in turn, the rotational modal projection and coordinate transformation equations are introduced to simulate the influence of the coupling between the two elements during the rotation of the gears; the meshing point of the gear pair changes continuously with the meshing operation, resulting in changes in the meshing point pair connected to the gear pair; in order to simulate the meshing of the bevel gear pair at any rotation angle, the "rotational modal projection" is introduced; for the driving wheel and the driven wheel, the corresponding modal projection matrix Vpi and V gi They are defined as the projection of the unit force on the modal coordinates of the driving wheel and the driven wheel respectively:

[0098] V pi =I i S p , V gj =I j S g i=1,2,3…z p j=1,2,3…z g

[0099] In the formula, z p and z g are the number of teeth of the driving wheel and the driven wheel respectively;

[0100] The polar angle of the i-th node and the polar angle of the j-th node are θ i and θ i Indicates; V pi and V gj Considered to be θ i and θ i The function is denoted as V p (θ i ) and V g (θ j ), whose expansions are expressed as:

[0101]

[0102] In the formula, v i a and v j a represents the projection of the unit force on a degree of freedom;

[0103] Therefore, the meshing stiffness matrix after polycondensation considering the change of meshing point is expressed as:

[0104]

[0105] The meshing stiffness matrix of the gear system after condensation considering the change of meshing point is expressed as:

[0106]

[0107] The corresponding meshing external force matrix is ​​expressed as:

[0108]

[0109] The external force vector considering the change of meshing point is expressed as:

[0110]

[0111] Therefore, the dynamic model of the gear system considering the change of meshing point during actual operation is obtained:

[0112]

[0113] The present invention also provides a computer-readable storage medium on which a computer program is stored, and the computer program implements the steps of the above method when executed by a processor.

[0114] Beneficial effects of the invention: The invention proposes a dynamic modeling method for a flexible bevel gear system that takes into account the change of meshing points, obtains the meshing stiffness matrix of the arc bevel gear contact, and obtains the dynamic equation of the flexible arc bevel gear system after dimension reduction. The proposed model greatly improves the solution efficiency while ensuring the solution accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] Figure 1 The present invention is a flow chart of the method for dynamic modeling of a flexible bevel gear system taking into account the change of meshing points.

[0116] Figure 2 The figure is a schematic diagram of the modeling of the flexible bevel gear model and the stiffness matrix assembly process;

[0117] Figure 3 It is a schematic diagram of the meshing unit of the flexible arc bevel gear pair;

[0118] Figure 4 Schematic diagram of force analysis of the spiral bevel gear system during operation; (a) meshing force of the driving wheel; (b) meshing force of the driven wheel;

[0119] Figure 5 Schematic diagram of the overall mass matrix and overall stiffness matrix of the dynamic model of the flexible bevel gear system; (a) overall mass matrix; (b) overall stiffness matrix;

[0120] Figure 6 Select the schematic diagram for the main node of the flexible bevel gear model;

[0121] Figure 7 The dynamic model of the bevel gear system considering the change of meshing point and the corresponding schematic diagram of the change of meshing point;

[0122] Figure 8 The figure is a comparison diagram of the modal frequency response function of the bevel gear in the free state and the frequency response function of the knocking modal test test result; (a) is the driving wheel frequency response function result obtained by the model of the present invention; (b) is the driving wheel frequency response function result of the test; (c) is the driven wheel frequency response function result obtained by the model of the present invention; (d) is the driven wheel frequency response function result of the test.

[0123] Fig. 9The figure is a comparison diagram of the pitch diameter modal vibration type of the driving and driven wheels of the model of the present invention and the finite element solid model; (a)-(e) are the model of the present invention; (f)-(j) are the finite element solid model;

[0124] Fig.10 It is a comparison chart between the simulation and experimental results of dynamic stress at the rim of the driven wheel under the traveling wave resonance speed; (a-1) and (a-2) are the time history and spectrum test results at the speed of gND1,1 respectively; (b-1) and (b-2) are the time history and spectrum simulation results at the speed of gND1,1 respectively; (c-1) and (c-2) are the time history and spectrum test results at the speed of gND1,2 respectively; (d-1) and (d-2) are the time history and spectrum simulation results at the speed of gND1,2 respectively;

[0125] Fig.11 Schematic diagram of dynamic stress distribution of the driven wheel matrix under traveling wave resonance speed; (a) is under gND1,1 resonance speed; (b) is under gND1,2 resonance speed.

[0126] Fig.12 Schematic diagram of the process of establishing the dynamic model of the flexible spiral bevel gear system considering the change of meshing point. DETAILED DESCRIPTION

[0127] In order to enable those skilled in the art to better understand the scheme of the present invention, the technical scheme in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present invention.

[0128] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0129] Combination Figure 1 As shown, the dynamic modeling method of the flexible bevel gear system considering the change of meshing point provided by the present invention is applied to the establishment of the dynamic model of the large and small gears of the spiral bevel gears meshing with each other. The specific steps of the method are as follows.

[0130] 1. Dynamic model of flexible spiral bevel gear system

[0131] The research on traveling wave resonance of spiral bevel gear system mainly focuses on analyzing the axial vibration of bevel gear matrix. Therefore, in the process of establishing the dynamic model of bevel gear system, the influence of gear matrix flexibility should be considered. In order to further analyze the vibration response of spiral bevel gear matrix along the axial direction, the dynamic model of spiral bevel gear considering the flexibility of gear matrix was established.

[0132] Considering the complex structure of the bevel gear matrix, a three-dimensional 8-node shell element is used to simulate the bevel gear matrix, and a Timoshenko beam element is used to simulate the complex shaft. The rigid connection between the inner hole of the gear matrix and the shaft is established through the interface coupling element. The construction process of the flexible bevel gear model and its stiffness matrix is ​​as follows: Figure 2 As shown. shell and n beam are the number of nodes for the shell model and the beam model, respectively.

[0133] The finite element modeling of the flexible gear matrix is ​​carried out using MATLAB software, and the unit mass matrix is ​​established. The unit mass matrix can be expressed as:

[0134]

[0135] Where N is the shape parameter matrix, ρ is the gear density, and J is the Jacobian matrix.

[0136] The element stiffness matrix can be expressed as:

[0137]

[0138] Where B is the strain matrix and D is the constitutive matrix.

[0139] The unit rotation softening matrix can be expressed as:

[0140]

[0141] Where Ω is the rotation matrix, which can be expressed as:

[0142]

[0143] In the formula, Ω x ,Ω y and Ω z are the components of the angular velocity in the x, y, and z directions.

[0144] The unit centrifugal stiffness matrix can be expressed as:

[0145]

[0146] Where, T c It can be expressed as:

[0147]

[0148] σ can be expressed as:

[0149]

[0150] Unit gyro matrix G b e It can be expressed as:

[0151]

[0152] In the formula, ψ, θ t and φ are the rotation angles along the x, y, and z directions, respectively.

[0153] The Timoshenko beam element is used to characterize the rotating shaft model, and its element mass matrix M s e , can be expressed as:

[0154]

[0155] In the formula, A S is the cross-sectional area of ​​each shaft unit, J is the polar area moment of inertia of each shaft unit, ρ is the material density, and l is the length of each shaft unit. a, b, c, d, f, and g are variables.

[0156] Element stiffness matrix K s e It can be expressed as:

[0157]

[0158] Where G is the shear modulus, E is the elastic modulus, and h, i, j, and k are as follows:

[0159]

[0160] Unit gyro matrix G s e It can be expressed as:

[0161]

[0162] Where p, q, s and w are as follows:

[0163]

[0164] The simulated gear matrix model is based on a three-dimensional 8-node shell element. Therefore, it is necessary to solve the time-varying meshing stiffness of the bevel gear pair under the root circle constraint state, and connect the nodes on the two gear matrix foundation columns through meshing springs, such as Figure 3 As shown, O s1 and O s2 is the connection node on the base column of the two gear bodies. Considering that the 8-node shell element contains translational and rotational degrees of freedom, on the basis of constraining the gear base circle and ignoring the rotational degrees of freedom generated during meshing, the O s1 and O s2The meshing spring between the two meshing planes does not need to project the torsional or translational stiffness excitation onto the meshing plane. The element stiffness matrix is ​​calculated using the projection method. Therefore, on this basis, the generalized displacement vector can be expressed as:

[0165] q m =[x p1 ,y p1 ,z p1 ,x g2 ,y g2 ,z g2 ]

[0166] The corresponding projection vector can be expressed as:

[0167]

[0168] In the formula, α b is the pressure angle, β b is the helix angle, δ p and δ g are the pitch angles of the driving wheel and the driven wheel respectively.

[0169] Therefore, the stiffness matrix of the meshing element under the constrained gear base circle state can be obtained:

[0170]

[0171] Where k(t) is the time-varying mesh stiffness (TVMS) of the spiral bevel gear pair.

[0172] In the process of stiffness matrix assembly, the stiffness matrix of the meshing element can be expressed as:

[0173]

[0174] During the meshing process of the bevel gear system, the tooth surface is subjected to normal meshing force. This force can be divided into radial, tangential and axial components. Therefore, the transmission system generates bending vibration, torsional vibration, axial vibration and torsional vibration caused by the axial dynamic meshing force.

[0175] Considering the stress of the spiral bevel gear system during the meshing process, the force analysis of the driving wheel and the driven wheel of the spiral bevel gear is carried out, and the normal meshing force of the gear is decomposed into component forces along each coordinate axis, such as Figure 4 shown.

[0176] During the operation of the gear system, vibration will cause strong nonlinear effects. The dynamic model of the flexible bevel gear system is usually only applicable to the case of no meshing loss. In the present invention, the gear torque is kept below 150N·m during the operation of the bevel gear system. Therefore, there is no meshing loss during the operation. Therefore, the present invention does not consider the influence of nonlinear meshing backlash. nProjected onto the x, y, and z axes respectively, we get the component force F of the driving wheel x1 、F y1 、F z1 and the component force F of the driven wheel x2 、F y2 、F z2 , which can be expressed as:

[0177]

[0178] The dynamic model of the flexible spiral bevel gear system can be expressed as:

[0179]

[0180] In the formula, ζ f is the overall displacement vector, F is the external load vector, M, C, G, and K are the overall mass matrix, overall damping matrix, overall gyro matrix, and overall stiffness matrix of the flexible spiral bevel gear system, respectively, which can be expressed as:

[0181] M=diag(M f_p ,M f_g )

[0182] G=diag(G f_p ,G f_g )

[0183]

[0184] Where M f_p 、M f_g , G f_p , G f_g is the overall mass matrix and overall gyro matrix of the driving wheel and the driven wheel, K f_p , K f_g is the overall stiffness matrix of the driving wheel and the driven wheel, which can be expressed as:

[0185]

[0186] In the formula, K p and K g They are the stiffness matrices of the driving wheel and the driven wheel respectively, and their grouping method is as follows Figure 2 As shown. σ_p and K σ_g Divided into the centrifugal stiffness matrix of the driving wheel and the driven wheel, K S_p and K S_g It is divided into the rotation softening matrix of the driving wheel and the driven wheel. C is the overall damping matrix, and the present invention adopts Rayleigh damping.

[0187] The stiffness matrix of the support bearing can be expressed as:

[0188] K b =diag(k bx ,k by ,k bz ,k bθx ,k bθy ,k bθz )

[0189] In the formula, k bx , k by ,,k bz ,,k bθx ,,k bθy and k bθz They represent the translation stiffness in the x, y, and z directions and the rotation stiffness around the x, y, and z axes respectively.

[0190] The support stiffness matrices of the driving wheel and the driven wheel can be expressed as:

[0191]

[0192] The overall mass matrix M is composed of the overall mass matrices of the driving wheel and the driven wheel, such as Figure 5 (a) is shown. The overall stiffness matrix K is composed of the gear base and shaft unit stiffness matrices of the active and driven bevel gears, meshing units, bearing units, and coupling units, as shown in Figure 5 (b) The stiffness matrices of the bevel gear base and the shaft are composed of the stiffness matrices of the three-dimensional 8-node shell element and the Timoshenko beam element, respectively. The gear base model and the shaft model are coupled by constraining the inner hole nodes of the bevel gear. The coupling process is shown in Figure 2 As shown in the figure, the coupling unit is established between the master / driven gear base and the shaft through a rigid connection between nodes, as shown in the figure. Figure 5 (b) as shown.

[0193] Two support bearings are respectively arranged on the driving wheel and the driven wheel shaft. The stiffness matrix of the support bearings is as follows: Figure 5 (b) shown. Where n g1 、n s1 、n g2 and n s2 The total number of nodes of the master / driven wheels and their corresponding rotating shafts are 448, 16, 860, and 16, respectively.

[0194] 2. Condensation of the dynamic model of the flexible gear system

[0195] Compared with the finite element model, the bevel gear model using shell elements and beam elements to simulate the tooth base and shaft respectively greatly improves the solution efficiency. On this basis, in order to further improve the calculation efficiency, the substructure modal synthesis (CMS) method is introduced. The physical coordinates are transformed into modal coordinates through the transformation matrix, which can be expressed as:

[0196] A=M or K

[0197] Where M and K are the overall mass matrix and stiffness matrix of the bevel gear system, respectively, and S is the transformation matrix. The superscript “-” indicates a condensed matrix.

[0198] S is composed of the main mode matrix Φ f and the constraint mode matrix Φ b Composition, can be expressed as:

[0199] S=[Φ f Φ b ],

[0200] The main modal matrix refers to the modal eigenvector when the main node is constrained. The constrained modal matrix refers to the displacement vector obtained by applying a unit displacement to the degree of freedom of a main node and constraining the degrees of freedom of all other main nodes, which can be expressed as:

[0201]

[0202] Where I is the unit matrix, V is the displacement vector from the node, and F is the external force.

[0203] like Figure 6 As shown in the figure, the base nodes and shaft segment nodes of the driving wheel and the driven wheel are selected as the main nodes of the bevel gear system. The number of main nodes is 32+16 and 43+16 respectively.

[0204] The vibration differential equation of the flexible bevel gear system model after polycondensation can be expressed as:

[0205]

[0206] The stiffness matrix of the dynamic model after polycondensation can be expressed as:

[0207]

[0208] In the formula, the subscripts "p" and "g" are the driving wheel and the driven wheel respectively, M f_p and M f_g They are the overall stiffness matrices of the flexible driving wheel and driven wheel of the bevel gear model respectively.

[0209] The overall rotation matrix of the kinetic model after polycondensation can be expressed as:

[0210]

[0211] In the formula, G f_p and G f_g are the overall rotation matrices of the bevel gear flexible driving wheel and driven wheel respectively.

[0212] After condensing the driving wheel and the driven wheel, the stiffness matrix of the meshing unit can be rewritten as:

[0213]

[0214] After the driving wheel and the driven wheel are condensed, the stiffness matrix of the supporting bearing can be rewritten as:

[0215]

[0216] The overall stiffness matrix of the dynamic model after polycondensation can be expressed as:

[0217]

[0218] In the formula, K f_p and K f_g They are the overall stiffness matrices of the flexible driving wheel and driven wheel of the bevel gear model respectively.

[0219] The damping matrix of the kinetic model after polycondensation can be expressed as:

[0220]

[0221] In the formula, C α and C β are the Rayleigh damping coefficients respectively.

[0222] The meshing external forces of the driving wheel and the driven wheel of the kinetic model after polycondensation can be expressed as:

[0223] f e =[f ep f eg ] T ,

[0224] The external load of the kinetic model after polycondensation can be expressed as:

[0225]

[0226] 3. Dynamic model of bevel gear system considering the change of meshing point

[0227] As the gears run, the meshing teeth are also changing. Existing research usually treats the meshing between gear pairs as equivalent to a constant meshing point connected by a spring. In order to simulate the actual meshing state between gear pairs more realistically, the changes of the meshing teeth involved in the meshing process are considered, and a dynamic model of the flexible arc bevel gear system is established that takes into account the changes in the meshing point during the actual rotation of the gears. The corresponding meshing point changes are as follows: Figure 7 shown. Figure 7 Middle pand z g are the number of teeth of the driving wheel and the driven wheel respectively.

[0228] When the driving wheel and the driven wheel are condensed, the number of main nodes of the gear matrix is ​​retained as the corresponding number of teeth. When the driving wheel and the driven wheel rotate and pass through each tooth node in turn, the rotational modal projection and coordinate transformation equations are introduced to simulate the influence of the coupling between the two elements during the gear rotation. The meshing point of the gear pair changes continuously with the meshing operation, which leads to the change of the meshing point pair connected to the gear pair. In order to simulate the meshing of the bevel gear pair at any rotation angle, "rotational modal projection" is introduced. For the driving wheel and the driven wheel, the corresponding modal projection matrix V pi and V gi They are defined as the projection of the unit force on the modal coordinates of the driving wheel and the driven wheel respectively:

[0229] V pi =I i S p , V gj =I j S g i=1,2,3…z p j=1,2,3…z g

[0230] In the formula, z p and z g are the number of teeth of the driving wheel and the driven wheel respectively.

[0231] The polar angle of the i-th node and the polar angle of the j-th node are represented by θ i and θ i Therefore, V pi and V gj It can be considered as θ i and θ i The function is denoted as V p (θ i ) and V g (θ j ), whose expansions can be expressed as:

[0232]

[0233] In the formula, v i a and v j a Represents the projection of the unit force on the a degree of freedom.

[0234] Therefore, the meshing stiffness matrix after polycondensation considering the change of meshing point can be expressed as:

[0235]

[0236] The meshing stiffness matrix of the gear system after condensation considering the change of meshing point can be expressed as:

[0237]

[0238] The corresponding meshing external force matrix can be expressed as:

[0239]

[0240] The external force vector considering the change of meshing point can be expressed as:

[0241]

[0242] Therefore, the dynamic model of the gear system considering the change of meshing point during actual operation can be obtained:

[0243]

[0244] Based on the above technical solution, the following combines the pitch diameter modal frequency and vibration shape simulation results of the finite element solid model of the bevel gear system, and uses a specific spiral bevel gear example as a verification case.

[0245] Table 1 Structural parameters of the example spiral bevel gear pair

[0246]

[0247] Table 2 Bearing support stiffness of example bevel gear system

[0248]

[0249] (1) The model of the present invention is based on the dynamic model of the flexible bevel gear system considering the changes of the meshing gear pair during the working process. The modal frequency response functions of the driving and driven wheels in the free state are compared with the frequency response functions of the knocking modal test results. Figure 8 (a)- Figure 8 As shown in (d), it can be seen that within the range of 8000 Hz, the driving wheel has two relatively obvious natural frequencies. Within the range of 6000 Hz, the driven wheel has three relatively obvious natural frequencies.

[0250] (2) The model of the present invention uses finite element solid units to establish the bevel gear model of the driving and driven wheels, and compares the modal calculation results of the model in the proposed method, the pitch diameter modal frequency calculation results of the finite element solid model with the test results, as shown in Table 3. The comparison results of the modal vibration shapes corresponding to the pitch diameter modal frequencies of the spiral bevel gears are shown in Table 3. Fig. 9 As shown. It can be seen that the second-order modal frequency of the driving wheel corresponds to two first-order pitch modes (defined as pND1,1 and pnd1,2), and their modal frequencies are defined as f pND1,1 and fpND1,2 The third-order modal frequencies of the gear correspond to two first pitch diameter modes (defined as gND1,1 and gND1,2) and one second pitch diameter mode (defined as gND2,1), and their modal frequencies are defined as f gND1,1 ,f gND1,2 and f gND2,1 The modal frequency results calculated by this method are compared with those calculated by the finite element solid model. The maximum error of the modal frequencies of the two gear pitch diameters is 3.6%, corresponding to f gND1,2 Pitch diameter vibration mode. The modal frequency results calculated by this method are compared with the test results. The maximum error of the pitch diameter modal frequencies of the two gears is f gND2,1 At the same time, the pitch modes of each order obtained by the proposed model are consistent with the pitch modes obtained by the finite element solid model, and the effectiveness of the model proposed in the present invention has been verified.

[0251] Table 3 Comparison of the pitch modal frequency of the model of the present invention with the finite element solid model and test results

[0252]

[0253] (3) The comparison results between the simulation and experimental test results of the dynamic stress response at the wheel rim at the traveling wave resonance speed in the gear working speed range of the model of the present invention are as follows: Fig.10 shown. Fig.10 (a-1), Fig.10 (a-2), Fig.10 (b-1) and Fig.10 (b-2) is the simulation and experimental test results of the gear vibration stress of the spiral bevel gear system running at the gND1,1 traveling wave resonance speed. At the gND1,1 traveling wave resonance speed, the meshing frequency f m The corresponding amplitude is less than f m +f r The corresponding amplitude. r is the rotation frequency of the shaft. The simulation results are consistent with the test results, indicating that the first radial wave resonance exists at this speed, proving the effectiveness of the proposed model. The corresponding gear foundation vibration stress distribution simulation results are shown in Fig.11 (a) It can be seen that at this resonant speed, the dynamic stress of the gear matrix presents the vibration form of the first pitch diameter. Fig.10 (c-1), Fig.10 (c-2), Fig.10 (d-1) and Fig.10 (d-2) It can be seen that at the gND1,2 traveling wave resonance speed, the f m -f r The corresponding amplitude is higher than f m On this basis, Fig.11The vibration stress distribution of the gear foundation in (b) is a 1-pitch diameter vibration mode, proving that the bevel gear system has a 1-pitch diameter wave resonance phenomenon at this speed.

[0254] The present invention has many specific application paths, and the above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be regarded as the protection scope of the present invention.

[0255] The serial numbers of the above embodiments of the present invention are only for description and do not represent the advantages and disadvantages of the embodiments. In the above embodiments of the present invention, the description of each embodiment has its own emphasis. For parts that are not described in detail in a certain embodiment, please refer to the relevant description of other embodiments. In the several embodiments provided in this application, it should be understood that the disclosed technical content can be implemented in other ways.

[0256] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A dynamic modeling method for a flexible bevel gear system considering the change of meshing point, characterized in that: The following steps are involved: Step 1: Based on the parameters of the spiral bevel gear, the structural parameters of the spiral bevel gear, the structural parameters of the driving wheel, and the structural parameters of the driven wheel, a three-dimensional 8-node shell element is used to simulate the bevel gear base, and a Timoshenko beam element is used to simulate the rotating shaft. A rigid connection between the inner hole of the bevel gear base and the rotating shaft is established through an interface coupling unit to obtain a finite element solid model of the bevel gear pair; Step 2: Based on the finite element solid model of the bevel gear pair, the time-varying meshing stiffness of the bevel gear pair is solved under the state of root circle constraint, and the nodes on the base columns of the two bevel gear bases are connected by meshing springs; Step 3: Consider the stress of the spiral bevel gear system during the meshing process, analyze the stress on the driving wheel and the driven wheel of the spiral bevel gear, and obtain the dynamic meshing force; Step 4: According to the time-varying meshing stiffness obtained in step 2 and the dynamic meshing force obtained in step 3, the dynamic model of the flexible spiral bevel gear system is obtained; Step 5: Based on the dynamic model of the flexible spiral bevel gear system obtained in step 4, the substructure modal synthesis method is introduced, and the physical coordinates are transformed into modal coordinates through the transformation matrix to obtain the vibration differential equation of the condensed flexible bevel gear system model; Step 6: Based on the vibration differential equation in step 5, the changes of the meshing teeth involved in the meshing process are considered, and a dynamic model of the flexible arc bevel gear system is established that considers the changes of the meshing point during the actual rotation of the gear. The rotational mode projection and coordinate transformation equations are introduced to simulate the influence of the gear rotation on the coupling between the two elements, and the dynamic model of the gear system that considers the changes of the meshing point during the actual operation is obtained.

2. A dynamic modeling method for a flexible bevel gear system considering the change of meshing point according to claim 1, characterized in that: The step 1 is specifically as follows: The finite element modeling of the flexible bevel gear matrix was carried out using MATLAB software, and a unit mass matrix was established; the unit mass matrix is ​​expressed as: Where N is the shape parameter matrix, ρ is the gear density, and J is the Jacobian matrix; The element stiffness matrix is ​​expressed as: Where B is the strain matrix and D is the constitutive matrix; Unit gyro matrix G b e It is expressed as: In the formula, ψ, θ t and φ are the rotation angles along the x, y, and z directions, respectively; The Timoshenko beam element is used to characterize the rotating shaft model, and its element mass matrix M s e It is expressed as: In the formula, A S is the cross-sectional area of ​​each shaft unit, J is the polar area moment of inertia of each shaft unit, ρ is the material density, l is the length of each shaft unit; a, b, c, d, f and g are variables; Element stiffness matrix K s e It is expressed as: Where G is the shear modulus, E is the elastic modulus, and h, i, j, and k are as follows: Where I is the diameter moment of inertia, φ is the shear influence factor; Unit gyro matrix G s e It is expressed as: Where Ω is the angular velocity of rotation, and p, q, s, and w are as follows: In the formula, 3. The method for dynamic modeling of a flexible bevel gear system considering the change of meshing point according to claim 1, characterized in that: The step 2 is specifically as follows: O s1 and O s2 is the connection node on the base column of the two bevel gears; considering that the 8-node shell element contains translational freedom and rotational freedom, on the basis of constraining the gear base circle, ignoring the rotational freedom generated during meshing, the O s1 and O s2 The meshing spring between the two meshing planes does not project the torsional stiffness or translational stiffness excitation onto the meshing plane. The unit stiffness matrix is ​​calculated using the projection method. On this basis, the generalized displacement vector is expressed as: q m =[x p1 ,y p1 ,z p1 ,x g2 ,y g2 ,z g2 ] x p1 Represents the x-direction freedom of the driving wheel; g2 represents the x-direction freedom of the driven wheel; y p1 Represents the y-direction freedom of the driving wheel; g2 Represents the y-direction freedom of the driven wheel; z p1 Represents the axial freedom of the driving wheel; z g2 Represents the axial freedom of the driven wheel; The corresponding projection vector is expressed as: In the formula, α b is the pressure angle, β b is the helix angle, δ p and δ g are the pitch cone angle of the driving wheel and the pitch cone angle of the driven wheel respectively; Therefore, the stiffness matrix of the bevel gear pair meshing unit under the constrained gear base circle state is obtained: Where k(t) is the time-varying mesh stiffness of the spiral bevel gear pair; In the process of stiffness matrix assembly, the stiffness matrix of the bevel gear pair meshing unit is expressed as:

4. The method for dynamic modeling of a flexible bevel gear system considering the change of meshing point according to claim 3 is characterized in that: In step 3, the normal meshing force of the gear is decomposed into component forces along each coordinate axis, specifically: During the operation of the bevel gear system, the gear torque is kept below 150 N·m. Therefore, the influence of nonlinear meshing backlash is not considered. The dynamic meshing force F n Projected onto the x, y, and z axes respectively, we get the component force F of the driving wheel x1 、F y1 、F z1 and the component force F of the driven wheel x2 、F y2 、F z2 , respectively expressed as:

5. The method for dynamic modeling of a flexible bevel gear system considering the change of meshing point according to claim 1, characterized in that: In step 4, the dynamic model of the flexible spiral bevel gear system is expressed as: In the formula, ζ f is the overall displacement vector, F is the external load vector, M, C, G, and K are the overall mass matrix, overall damping matrix, overall gyro matrix, and overall stiffness matrix of the flexible spiral bevel gear system, respectively, which are expressed as: M=diag(M f_p ,M f_g ) G=diag(G f_p ,G f_g ) Where M f_p 、M f_g are the overall mass matrices of the driving wheel and the driven wheel respectively; G f_p , G f_g are the overall gyro matrices of the driving wheel and the driven wheel, respectively, K f_p , K f_g is the overall stiffness matrix of the driving wheel and the driven wheel, expressed as; In the formula, K p and K g are the stiffness matrices of the driving wheel and the driven wheel respectively; K σ_p and K σ_g are the centrifugal stiffness matrices of the driving wheel and the driven wheel, K S_p and K S_g are the rotation softening matrices of the driving wheel and the driven wheel respectively; C is the overall damping matrix; The stiffness matrix of the support bearing is expressed as: K b =diag(k bx ,k by ,k bz ,k bθx ,k bθy ,k bθz ) In the formula, k bx , k by ,,k bz ,,k bθx ,,k bθy and k bθz Respectively represent the translation stiffness in the x, y, z directions and the rotation stiffness around the x, y, z axes; The support stiffness matrices of the driving wheel and the driven wheel are expressed as: The overall mass matrix M is composed of the overall mass matrix of the driving wheel and the driven wheel; the overall stiffness matrix K is composed of the stiffness matrix of the gear matrix and the shaft unit of the driving wheel and the driven wheel, the bevel gear pair meshing unit, the bearing unit and the coupling unit; the stiffness matrix of the bevel gear matrix and the shaft is composed of the stiffness matrix of the three-dimensional 8-node shell unit and the Timoshenko beam unit respectively. The gear matrix model and the shaft model are coupled by constraining the inner hole nodes of the bevel gear. The coupling unit is established between the driving / driven gear matrix and the shaft through the rigid connection between the nodes; Two supporting bearings are respectively arranged on the driving wheel and the driven wheel rotating shaft.

6. The method for dynamic modeling of a flexible bevel gear system considering the change of meshing point according to claim 1, characterized in that: Step 5 is as follows: Where, M and K are the overall mass matrix and stiffness matrix of the bevel gear system, respectively, S is the transformation matrix; the superscript "-" indicates the condensation matrix; S is composed of the main mode matrix Φ f and the constraint mode matrix Φ b Composition, expressed as: The main modal matrix refers to the modal eigenvector when the main node is constrained; the constrained modal matrix refers to the displacement vector obtained by applying a unit displacement to the degree of freedom of a main node and constraining the degrees of freedom of all other main nodes, expressed as: Where I is the unit matrix, V is the displacement vector from the node, and F is the external force; Select the bevel gear base nodes and shaft segment nodes of the driving wheel and the driven wheel as the main nodes of the bevel gear system respectively; The vibration differential equation of the flexible bevel gear system model after polycondensation is expressed as: In the formula, They are the overall mass matrix, overall damping matrix, overall gyro matrix and overall stiffness matrix of the flexible spiral bevel gear system after polycondensation. is the external load vector after polycondensation; The stiffness matrix of the dynamic model after polycondensation is expressed as: In the formula, the subscripts "p" and "g" are the driving wheel and the driven wheel respectively, M f_p and M f_g They are the overall stiffness matrices of the flexible driving wheel and driven wheel of the bevel gear model respectively; The overall rotation matrix of the kinetic model after polycondensation is expressed as: In the formula, G f_p and G f_g are the overall rotation matrices of the bevel gear flexible driving wheel and driven wheel respectively; After the driving wheel and the driven wheel are condensed, the stiffness matrix of the bevel gear pair meshing unit is rewritten as: After the driving wheel and the driven wheel are condensed, the stiffness matrix of the supporting bearing is rewritten as: The overall stiffness matrix of the dynamic model after polycondensation is expressed as: In the formula, K f_p and K f_g They are the overall stiffness matrices of the flexible driving wheel and driven wheel of the bevel gear model respectively; The damping matrix of the kinetic model after polycondensation is expressed as: In the formula, C α and C β are the Rayleigh damping coefficients, respectively; The meshing external forces of the driving wheel and the driven wheel of the kinetic model after polycondensation are expressed as: f ep 、f eg They are the meshing external forces of the driving wheel and the driven wheel of the flexible spiral bevel gear system, is the meshing external force of the driving wheel and the driven wheel after condensation; The external load of the kinetic model after polycondensation is expressed as:

7. The method for dynamic modeling of a flexible bevel gear system considering the change of meshing point according to claim 1, characterized in that: In step 6, When the driving wheel and the driven wheel are condensed, the number of main nodes of the gear matrix is ​​retained as the number of teeth corresponding to them. When the driving wheel and the driven wheel rotate and pass through each tooth node in turn, the rotational mode projection and coordinate transformation equations are introduced to simulate the influence of the coupling between the two elements during the gear rotation process; the meshing point of the gear pair changes continuously with the meshing operation, which leads to the change of the meshing point pair connected to the gear pair; in order to simulate the meshing of the bevel gear pair at any rotation angle, the "rotational mode projection" is introduced; for the driving wheel and the driven wheel, the corresponding mode projection matrix V pi and V gi They are defined as the projection of the unit force on the modal coordinates of the driving wheel and the driven wheel respectively: V pi =I i S p ,V gj =I j S g i=1,2,3…z p j=1,2,3…z g In the formula, z p and z g are the number of teeth of the driving wheel and the driven wheel respectively; The polar angle of the i-th node and the polar angle of the j-th node are θ i and θ i Indicates; V pi and V gj Considered to be θ i and θ i The function is denoted as V p (θ i ) and V g (θ j ), whose expansions are expressed as: In the formula, v i a and v j a represents the projection of the unit force on a degree of freedom; Therefore, the meshing stiffness matrix after polycondensation considering the change of meshing point is expressed as: The meshing stiffness matrix of the gear system after condensation considering the change of meshing point is expressed as: The corresponding meshing external force matrix is ​​expressed as: The external force vector considering the change of meshing point is expressed as: Therefore, the dynamic model of the gear system considering the change of meshing point during actual operation is obtained:

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