CN117807721A - Kinetic modeling method considering change of meshing point of bevel gear - Google Patents
Kinetic modeling method considering change of meshing point of bevel gear Download PDFInfo
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Abstract
The invention discloses a dynamic modeling method considering change of meshing points of bevel gears, which comprises the following steps: s1, modeling a bevel gear and a transmission shaft; s2, assembling a master bevel gear shaft system and a slave bevel gear shaft system; s3, modeling of meshing pairs: in the dynamics model, a meshing unit is established between two bevel gears, the meshing process is simulated as the mutual influence of meshing force of equivalent meshing points of the two gears through springs, the equivalent meshing points are rigidly coupled with tooth surfaces meshed at the same moment, and corresponding input torque and output load are respectively applied to the input and output positions of the two transmission shafts. According to the invention, the equivalent meshing points are projected onto the gear shaft line, the position of each meshing point is read to calculate the meshing force arm, dynamic modeling of the meshing points is realized through the change of the meshing force arm, and the modeling precision is improved.
Description
Technical Field
The invention relates to the technical field of structural design of gear transmission systems, in particular to a dynamic modeling method considering change of meshing points of bevel gears.
Background
The transmission system adopts bevel gears to transmit non-parallel torque or speed, and the gear dynamics model is a mechanical model for researching the gear transmission system. Gears are a common mechanical element and are widely used in transmissions in various fields. By studying the dynamic characteristics of the gears, the performance of the transmission system can be evaluated and optimized. The existing bevel gear dynamics model simplifies a spiral bevel gear into a rigid body, and meshing force is applied to tooth surfaces, but because the gear is a rigid body, the force is equivalent to the force applied to a node of a shaft where the gear is positioned, and only the torque in all directions is applied to the node of the shaft besides the force in all directions. However, the engagement point is changed by the torque and the engagement process during the engagement of the bevel gears, resulting in a change in the arm of force. Larger errors can be introduced if the engagement point variation is not considered.
Disclosure of Invention
The present invention aims to solve at least one of the technical problems existing in the prior art. Therefore, the invention provides a dynamic modeling method considering the change of the meshing point of the bevel gear, which can simplify the calculation of the change of the moment arm of the meshing point and improve the modeling precision.
According to the embodiment of the invention, the dynamic modeling method for considering the change of the meshing point of the bevel gear comprises the following steps:
s1, modeling a bevel gear and a transmission shaft: modeling a bevel gear by adopting a 20-node hexahedral unit, performing dimension-reducing polycondensation on the bevel gear by adopting a mode synthesis method so as to reduce the size of a matrix, and modeling a transmission shaft by adopting a Timoshenko beam unit;
s2, assembling a master bevel gear shaft system and a slave bevel gear shaft system: coupling the bevel gear and the drive shaft through a coupling unit, and applying rigidity of the bearing to a bearing node on the drive shaft to simulate the bearing;
s3, modeling of meshing pairs: in a dynamics model, a meshing unit is established between two bevel gears, a meshing process is simulated as the mutual influence of meshing force of equivalent meshing points of the two gears through springs, the equivalent meshing points are coupled with tooth surfaces meshed at the same moment, corresponding input torque and output load are respectively applied to input and output positions of the two transmission shafts, the equivalent meshing points are projected onto the axes of the bevel gears, and a rigidity matrix of the meshing unit is expressed as:
K m ＝k m (t)(V) T V，
wherein k is m (t) is a time-varying engagement stiffness obtained by load contact analysis, V represents an engagement vector,
V＝[V g V p ]，
V i ＝[n ix n iy n iz λ ix λ iy λ iz ]，
n i ＝(n ix n iy n iz ) T represents the unit component of the meshing force in the direction X, Y, Z, lambda ix 、λ iy And lambda (lambda) iz Is the equivalent moment arm component of the moment arm in the X, Y, Z direction;
wherein the equivalent meshing point is the center of the resultant force on the tooth surface, and the direction points to the direction of the resultant force.
According to some embodiments of the invention, in said step S3 λ ix ＝n i ·(u xi ×r i )，λ iy ＝n i ·(u yi ×r i )，λ iz ＝n i ·(u zi ×r i )，n i And r i Are all obtained by gear loading contact analysis, u xi ＝(1 0 0) T ,u yi ＝(0 1 0) T ,u zi ＝(0 0 1) T 。
According to some embodiments of the invention, in said step S3, r i ＝(r ix r iy r iz ) T ，r ix 、r iy 、r iz Is the coordinate component of the engagement point at X, Y, Z.
According to some embodiments of the invention, step S1 comprises the steps of:
s101, modeling the bevel gear by adopting the 20-node hexahedral unit, dividing a plurality of 20-node hexahedral units according to the structure of the bevel gear, and assembling the units according to the unit numbers of the 20-node hexahedral units to obtain a total mass matrix and a rigidity matrix of the bevel gear, wherein the mass matrix of one 20-node hexahedral unit is as follows:
the stiffness matrix of one 20-node hexahedral unit is:
wherein N represents a shape function, ρ 1 Is the density of bevel gears, B and B * The strain matrix under the global coordinate system and the iso-coordinate system respectively, D is an elastic matrix, W i 、W j 、W k For Gaussian weight, τ is the number of Gaussian points in each direction, J is the Jacobian matrix, and ζ, η, ζ are the integral point coordinates.
According to some embodiments of the invention, the shape function N is expressed as:
according to some embodiments of the invention, in the shape function N, for an angular node, the shape function is expressed as:
according to some embodiments of the invention, in the shape function N, for the intermediate node, the shape function is expressed as:
wherein, xi a 、η a 、ζ a The values of (a) are selected according to the position of each node in the iso-coordinate system,the subscript a indicates a node on a 20-node hexahedral unit.
According to some embodiments of the invention, the stiffness matrix of the coupling unit of the bevel gear and the drive shaft is expressed as:
wherein K is C A diagonal matrix of 6*6, the diagonal value of which is an equally large number.
According to some embodiments of the invention, the master and slave bevel gears are mounted at 90 degrees in the kinetic model.
According to the embodiment of the invention, the dynamic modeling method considering the change of the meshing point of the bevel gear has at least the following beneficial effects:
(1) The meshing point is directly projected onto the shaft, the coordinate system established by the equivalent meshing point is the same as the contact analysis coordinate system, the position of each meshing point, namely the value in the meshing process, is directly read, and then the equivalent force arm is calculated, so that the method is simple and is not easy to make mistakes;
(2) The modeling considers the change of engagement points in different engagement processes under different torques; the more realistic vibration characteristics are obtained;
(3) The flexible spiral bevel gear does not need to repeatedly adopt a mode synthesis method to reduce the dimension of the matrix, only needs to perform condensation polymerization calculation once, and effectively improves the calculation speed.
Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
Drawings
The invention is further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a model diagram of a bevel gear drive system as established herein;
FIG. 2 is a schematic illustration of the meshing point established near the gear contact surface and stationary;
FIG. 3 is a schematic view of the engagement point projected onto the shaft;
FIG. 4 is a schematic illustration of the change in the engagement point direction vector in the X direction over an engagement period;
FIG. 5 is a schematic illustration of the variation of the engagement point direction vector in the Y direction over an engagement period;
FIG. 6 is a schematic illustration of the change in the engagement point direction vector in the Z direction over an engagement period;
FIG. 7 is a schematic illustration of the change in engagement point in the X direction over an engagement period;
FIG. 8 is a schematic illustration of the variation of engagement points in the Y direction over an engagement period;
FIG. 9 is a schematic illustration of the change in engagement point in the Z direction over an engagement period;
FIG. 10 is a schematic illustration of the effect of torque magnitude on the average engagement point in the X direction;
FIG. 11 is a schematic illustration of the effect of torque magnitude on the average engagement point in the Y-direction;
FIG. 12 is a schematic illustration of the effect of torque magnitude on the average engagement point in the Z-direction;
FIG. 13 is a time domain plot of dynamic transmission errors for fixed and time varying points;
FIG. 14 is a graph of a fixed meshing point spectrum;
FIG. 15 is a graph of a time-varying meshing point spectrum;
FIG. 16 is a graph of dynamic drive error sweeps for different torque.
Detailed Description
Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to like or similar elements or elements having like or similar functions throughout. The embodiments described below by referring to the drawings are illustrative only and are not to be construed as limiting the invention.
In the description of the present invention, it should be understood that the direction or positional relationship indicated with respect to the description of the orientation, such as up, down, etc., is based on the direction or positional relationship shown in the drawings, is merely for convenience of describing the present invention and simplifying the description, and does not indicate or imply that the apparatus or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus should not be construed as limiting the present invention.
In the description of the present invention, plural means two or more. The description of the first and second is for the purpose of distinguishing between technical features only and should not be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated or implicitly indicating the precedence of the technical features indicated.
In the description of the present invention, unless explicitly defined otherwise, terms such as arrangement, installation, connection, etc. should be construed broadly and the specific meaning of the terms in the present invention can be reasonably determined by a person skilled in the art in combination with the specific contents of the technical scheme.
According to the dynamic modeling method considering the change of the meshing point of the bevel gear, a 20-node hexahedral unit is adopted to establish the bevel gear, the bevel gear is subjected to dimension reduction polycondensation by a modal synthesis method so as to improve the calculation speed, a Timoshenko beam unit is adopted to establish a transmission shaft, then the transmission shaft and the bevel gear are combined through a coupling unit, the meshing tooth surface is coupled to an equivalent meshing point, finally vibration characteristic analysis of a bevel gear transmission system is carried out, and a vibration displacement cloud picture under the resonance rotation speed is drawn to provide theoretical guidance for the optimization design of the structure, wherein the method specifically comprises the following steps:
s1, modeling a bevel gear and a transmission shaft: carrying out flexible modeling on the bevel gear by adopting a 20-node hexahedral unit to obtain a quality matrix and a rigidity matrix of the bevel gear, fully simulating the inherent attribute of the bevel gear which is a complex structure, carrying out dimension-reducing polycondensation on the established bevel gear by adopting a mode synthesis method to reduce the size of the matrix, and simultaneously carrying out modeling on the transmission shaft by adopting a Timoshenko beam unit; the bevel gear drive system model is shown in fig. 1.
S2, assembling a bevel gear shaft system: coupling a bevel gear and a transmission shaft through a coupling unit, coupling a coupling node of an inner ring of the bevel gear and a shaft node of the transmission shaft where the bevel gear is positioned, obviously, coupling a driving gear with a driving shaft, coupling a driven gear with a driven shaft, and then applying rigidity of a bearing on a bearing node of the transmission shaft to simulate the bearing;
s3, modeling of meshing pairs: in the dynamics model, an engagement unit is established between equivalent engagement points of two bevel gears (a driving gear and a driven gear), the engagement process is simplified into a time-varying spring, and the stiffness matrix of the engagement unit can be expressed as:
K m ＝k m (t)(V) T V，
k m (t) is a time-varying engagement stiffness, obtained by load contact analysis, and the engagement vector V is expressed as:
V＝[V g V p ]，
V i ＝[n ix n iy n iz λ ix λ iy λ iz ]，
n i ＝(n ix n iy n iz ) T represents the unit component of the meshing force in the direction X, Y, Z, lambda ix 、λ iy And lambda (lambda) iz Is the equivalent moment arm component of the moment arm in the direction of X, Y, Z.
The method comprises the steps that force interaction is carried out on equivalent meshing points of two bevel gears through springs, the equivalent meshing points are rigidly coupled with tooth surfaces meshed at the same moment, corresponding input torque and corresponding output load are respectively applied to input and output positions (a driving shaft and a driven shaft) of the two transmission shafts, wherein the equivalent meshing points are centers of resultant forces on the tooth surfaces, and the directions of the equivalent meshing points point to directions of the resultant forces; and modeling of the bevel gear transmission system is completed after the three steps are carried out. Since the spiral bevel gear is a precise transmission part, the meshing area of the tooth surfaces is constantly changed, and if the meshing point is regarded as a fixed point, the meshing point is inaccurate. Whereas the prior art bevel gear engagement points are generally fixed, a model of the fixed bevel gear engagement points is shown in fig. 2. If the meshing point is established near the gear contact surface, it is stationary. Then V i Rewritten as
V i ＝[n ix n iy n iz 0 0 0]，
Because the meshing point of the established meshing coordinate system is the origin, lambda ix 、λ iy And lambda (lambda) iz And are all 0. Although the establishment of the meshing point in the vicinity of the gear contact surface may take into account the variation of the meshing point, this is very complex, since the variation of the meshing point is based on a contact analysis coordinate system, which requires a complicated conversion of the position on the contact analysis coordinate system to the meshing coordinate system. The model of the change of the meshing point of the bevel gear used in the scheme is shown in fig. 3, if the meshing point is directly projected onto the shaft, the coordinate system established by the equivalent meshing point is the same as the contact analysis coordinate system, the value of each meshing point position in the meshing process is directly read, and then the equivalent force arm is calculated, so that the method is simple and is not easy to make mistakes.
In some embodiments of the invention, step S1 comprises the steps of:
s101, modeling a bevel gear by adopting 20-node hexahedral units, dividing a plurality of 20-node hexahedral units according to the structure of the bevel gear, and assembling the units according to the unit numbers of the 20-node hexahedral units to obtain the overall quality matrix and the stiffness matrix of the bevel gear, wherein the quality matrix of one 20-node hexahedral unit is as follows:
the stiffness matrix of a 20-node hexahedral cell is:
wherein N represents a shape function, ρ 1 Is the density of bevel gears, B and B * The strain matrix under the global coordinate system and the iso-coordinate system respectively, D is an elastic matrix, W i 、W j 、W k For Gaussian weight, τ is the number of Gaussian points in each direction, J is the Jacobian matrix, and ζ, η, ζ are the integral point coordinates.
In some embodiments of the invention, the shape function N is expressed as:
as shown in fig. 2, in some embodiments of the present invention, for the corner nodes of a hexahedral cell, its shape function is expressed as:
in some embodiments of the invention, for the intermediate node of the hexahedral cell, its shape function is expressed as:
in xi a 、η a 、ζ a The values of (a) are selected according to the positions of the nodes in an iso-reference coordinate system, and a subscript a represents a node on a 20-node hexahedral unit; it should be noted that a 20-node hexahedral unit refers to a hexahedral unit having 20 nodes, each having three degrees of freedom.
In some embodiments of the invention, λ in said step S3 ix ＝n i ·(u xi ×r i )，λ iy ＝n i ·(u yi ×r i )，λ iz ＝n i ·(u zi ×r i )，n i And r i Are all obtained by gear loading contact analysis, u xi ＝(1 0 0) T ,u yi ＝(0 1 0) T ,u zi ＝(0 0 1) T 。
In some embodiments of the invention, in said step S3, r i ＝(r ix r iy r iz ) T ，r ix 、r iy 、r iz Is the coordinate component of the engagement point at X, Y, Z. It is envisioned that the constant pattern of bevel gear mesh points is shown in fig. 2, if the mesh points are established near the gear contact surface and are stationary. Then V i Rewritten as
V i ＝[n ix n iy n iz 0 0 0]，
Because the meshing point of the established meshing coordinate system is the origin, r i ＝(r ix r iy r iz ) T The values of the inner parts are all 0, lambda ix 、λ iy And lambda (lambda) iz And are all 0. Although the establishment of the meshing point in the vicinity of the gear contact surface may take into account the variation of the meshing point, this is very complex, since the variation of the meshing point is based on a contact analysis coordinate system, which requires a complicated conversion of the position on the contact analysis coordinate system to the meshing coordinate system. Such as r when the engagement point is the coupling point i ＝(r ix r iy r iz ) T The inner values are all 0, and the next time needs to calculate the current r i The positional relationship between the engagement point and the next engagement point is relatively complicated and is prone to error. The model of the change of the meshing point of the bevel gear used in the scheme is shown in fig. 3, if the meshing point is directly projected on the shaft, the coordinate system established by the equivalent meshing point is the same as the contact analysis coordinate system, and the position of each meshing point, namely r, is directly read i ＝(r ix r iy r iz ) T The value during engagement is then calculated as the equivalent moment arm, which is simple and less prone to error.
In some embodiments of the present invention, step S1 further includes step S102 of obtaining the transformation matrix T by using a mode synthesis method cms Using a transformation matrix T cms The following transformations were performed:
M cms ＝T cms T MT cms ，
K cms ＝T cms T KT cms ，
obtaining a quality matrix M after bevel gear polycondensation according to the above cms And a stiffness matrix K cms Realizing dimension reduction polycondensation of the built bevel gear; wherein M and K are the mass matrix and the rigidity matrix of the bevel gear overall; specifically, a mode synthesis method is adopted to obtain a conversion matrix T cms The method comprises the following steps:
firstly, according to the characteristics of bevel gears, each bevel gear is provided with two interfaces, one is the meshing tooth surface of the bevel gear, and the other is the inner ring surface of the bevel gear connected with a transmission shaft; establishing a motion equation of a substructure of the bevel gear under physical coordinates by using node displacement:
wherein C is a damping matrix, Q represents an external load vector, R represents a force vector at the interface, a represents a node displacement vector, and a is divided into internal displacements a i And interface displacement a j The two parts, Q and R can be divided into two parts, and the motion equation of the substructure can be written as:
the free equation of motion without damping is written as:
wherein M is ii 、K ii Is an internal mass matrix and a stiffness matrix, M jj 、K jj Is the mass matrix and stiffness matrix at the interface, M ij And M ji Is a coupling quality matrix, K ij And K ji Is a coupling stiffness matrix, R j Is the force vector at the interface;
the free equation of motion without damping is then calculated in modal coordinates:
the interface is fixed firstly, namely, the a is made j =0, calculating the natural mode of the subsystem at the fixed interface, i.e. solving the following free vibration equation eigenvalues:
and then, solving the eigenvalue of the free vibration equation to obtain i natural vibration modes, wherein the combination is represented as phi by a matrix n The method comprises the steps of carrying out a first treatment on the surface of the Incidentally, Φ n The following operations can be performed:
wherein I is i Is an identity matrix with dimension i.i., omega ii A matrix with main diagonal as characteristic value;
the interface has been fixed above, after which each degree of freedom at the interface is released in turn and then the static displacement is calculated:
from the above formula:
a i ＝-K ii -1 K ij a j ，
let a j The j elements in the model are sequentially taken as unit values, the rest are 0, and the corresponding j groups of static displacement vectors, namely constraint modes, a, are obtained i The form combined into matrix is expressed as phi j ：
Φ j ＝-K ii -1 K ij I j ＝-K ii -1 K ij ,
Meanwhile, to achieve the purpose of reducing the degree of freedom, phi is omitted n Only the k rows of low-order main modes are reserved to form phi k The following operation is performed to obtain a transformation matrix T cms ：
Wherein I is j Is an identity matrix with dimension j x j.
If the flexible spiral bevel gear modeling is adopted, and the spiral bevel gears in different meshing states need to be subjected to matrix dimension reduction polycondensation by adopting a mode synthesis method in consideration of the change of meshing points, three coordinate systems (a contact analysis coordinate system, a previous meshing point coordinate system and a next meshing point coordinate system) are involved in very time consumption or very complex coordinate change of the meshing points. But projecting the engagement point on the axis does not require repeated dimension reduction and modifying the position of the engagement point is very simple.
In some embodiments of the present invention, step S1 further includes step S103: carrying out flexible modeling on the transmission shaft by adopting a Timoshenko beam unit, dividing a plurality of Timoshenko beam units according to the structure of the transmission shaft, and carrying out unit assembly on each unit according to the unit numbers of the Timoshenko beam units so as to obtain the overall mass matrix and the stiffness matrix of the transmission shaft; as shown in fig. 3, a beam unit is composed of two nodes, each node has 6 degrees of freedom, and the displacement array of the units is:
q s ＝[x i ,y i ,z i ,θ xi ,θ yi ,θ zi ,x (i+1) ,y (i+1) ,z (i+1) ,θ x(i+1) ,θ y(i+1) ,θ z(i+1) ] T ，
wherein x is i 、y i 、z i Is the translational degree of freedom, theta xi 、θ yi 、θ zi Is a degree of freedom of rotation;
the displacement expression of any point in the unit in each direction is:
z＝N d q s ，
θ z ＝N r q s ，
wherein,
N d ＝[0 0 N d1 0 0 0 0 0 N d1 0 0 0]，
N r ＝[0 0 0 0 0 N r1 0 0 0 0 0 N r2 ]，
through the formula, the mass matrix of the Timoshenko beam unit is calculated by using a Lagrange equation and a kinetic energy formula, the rigidity matrix of the Timoshenko beam unit is derived by using the Lagrange equation and a potential energy formula, and the mass matrix of one Timoshenko beam unit is expressed as follows:
the stiffness matrix of a Timoshenko beam unit is expressed as:
wherein,
in the above formula ρ 2 Is the density of the transmission shaft, v is Poisson's ratio, E is an elastic model, m is the ratio of the inner diameter to the outer diameter of the beam unit, mu is a hollow correction coefficient, G is the shear modulus of the beam unit, A is the cross-sectional area of the beam unit, l is the length of the beam unit, I s 、I p The diameter moment of inertia and the pole moment of inertia of the beam unit respectively,to simplify the coefficients.
In the embodiment, a 20-node hexahedral unit is adopted to build a model of the bevel gear, so that the complex structure of the bevel gear can be accurately simulated, a Timoshenko beam unit is adopted to build a model of the transmission shaft, and the influence of parameters of all shafts such as the outer diameter, the inner diameter and the length of different shafts on the modal frequency can be rapidly analyzed.
In some embodiments of the present invention, referring to fig. 4, in step S2, the bevel gear and the drive shaft are coupled by the coupling unit, and the coupling node of the bevel gear inner race and the shaft node of the drive shaft where the bevel gear is located are coupled, and the stiffness matrix of the coupling unit of the bevel gear and the drive shaft is expressed as:
wherein K is C A diagonal matrix of 6*6, the diagonal value of which is an equally large number.
Implementation case: the moment arm of the spiral bevel gear changes periodically along with the meshing process when 500Nm of torque is transmitted, and the meshing point direction vector changes in a meshing period in the X, Y, Z direction as shown in fig. 4, 5 and 6 respectively.
As shown in fig. 7, 8 and 9, the moment arm in the direction X, Y, Z changes as the meshing progresses, because the contact area of the spiral bevel gear changes greatly in one meshing cycle, so that the equivalent meshing also changes, and if an average moment arm is adopted, the dynamics of the spiral bevel gear has errors. The meshing parameters of the moment arms of the spiral bevel gears when the output torque is 500, 1000 and 2000Nm are shown in fig. 10, 11 and 12, the average moment arm of the spiral bevel gears can also change to different degrees along with the change of the torque, and the contact areas of the spiral bevel gears with different torques can also change the equivalent meshing points due to the deflection of the contact areas caused by the deformation. The time domain diagram of the dynamic transmission error between the fixed point and the time-varying point is shown in fig. 13, and it can be seen that the time-varying engagement point and the fixed engagement point differ in the variation and the amplitude of the dynamic transmission error. The frequency spectra of the time-varying engagement points and the fixed engagement points are shown in fig. 14 and 15. It can also be seen that the spectral composition of the fixed engagement point differs from the time-varying engagement point, and that ignoring the variation in engagement point during engagement may be inaccurate.
Fig. 16 is a graph of dynamic transmission error sweeps at different torque, and it can be found that the dynamic transmission error standard deviation is shifted upwards with increasing torque, and the resonance rotation speed is changed due to the change of the position and direction of the engagement point at different torque. The change in position and direction of the engagement point needs to be considered for different torques and engagement processes.
The embodiments of the present invention have been described in detail with reference to the accompanying drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of one of ordinary skill in the art without departing from the spirit of the present invention.
Claims (9)
1. The dynamic modeling method taking the change of the meshing point of the bevel gear into consideration is characterized by comprising the following steps of:
s1, modeling a bevel gear and a transmission shaft: modeling a bevel gear by adopting a 20-node hexahedral unit, performing dimension-reducing polycondensation on the bevel gear by adopting a mode synthesis method so as to reduce the size of a matrix, and modeling a transmission shaft by adopting a Timoshenko beam unit;
s2, assembling a master bevel gear shaft system and a slave bevel gear shaft system: coupling the bevel gear and the drive shaft through a coupling unit, and applying rigidity of the bearing to a bearing node on the drive shaft to simulate the bearing;
s3, modeling of meshing pairs: in a dynamics model, a meshing unit is established between two bevel gears, a meshing process is simulated as the mutual influence of meshing force of equivalent meshing points of the two gears through springs, the equivalent meshing points are coupled with tooth surfaces meshed at the same moment, corresponding input torque and output load are respectively applied to input and output positions of the two transmission shafts, the equivalent meshing points are projected onto the axes of the bevel gears, and a rigidity matrix of the meshing unit is expressed as:
K m ＝k m (t)(V) T V，
wherein k is m (t) is a time-varying engagement stiffness obtained by load contact analysis, V represents an engagement vector,
V＝[V g V p ]，
V i ＝[n ix n iy n iz λ ix λ iy λ iz ]，
n i ＝(n ix n iy n iz ) T represents the unit component of the meshing force in the direction X, Y, Z, lambda ix 、λ iy And lambda (lambda) iz As an equivalent moment arm component of the moment arm in the direction X, Y, Z,
wherein the equivalent meshing point is the center of the resultant force on the tooth surface, and the direction points to the direction of the resultant force.
2. The kinetic modeling method considering the change of the meshing point of the bevel gear according to claim 1, wherein: in step S3, lambda ix ＝n i ·(u xi ×r i )，λ iy ＝n i ·(u yi ×r i )，λ iz ＝n i ·(u zi ×r i )，n i And r i Are all obtained by gear loading contact analysis, u xi ＝(1 0 0) T ,u yi ＝(0 1 0) T ,u zi ＝(0 0 1) T 。
3. The kinetic modeling method considering the change of the meshing point of the bevel gear according to claim 2, wherein: in step S3, r i ＝(r ix r iy r iz ) T ，r ix 、r iy 、r iz Is the coordinate component of the engagement point at X, Y, Z.
4. The kinetic modeling method considering the change of the meshing point of the bevel gears according to claim 1, wherein: step S1 comprises the steps of:
s101, modeling the bevel gear by adopting the 20-node hexahedral unit, dividing a plurality of 20-node hexahedral units according to the structure of the bevel gear, and assembling the units according to the unit numbers of the 20-node hexahedral units to obtain a total mass matrix and a rigidity matrix of the bevel gear, wherein the mass matrix of one 20-node hexahedral unit is as follows:
the stiffness matrix of one 20-node hexahedral unit is:
wherein N represents a shape function, ρ 1 Is the density of bevel gears, B and B * The strain matrix under the global coordinate system and the iso-coordinate system respectively, D is an elastic matrix, W i 、W j 、W k Is high enough toThe Gaussian weight, τ is the number of Gaussian points in each direction, J is the Jacobian matrix, and ζ, η and ζ are the integral point coordinates.
5. The kinetic modeling method considering the change of the meshing point of the bevel gear according to claim 4, wherein: the shape function N is expressed as:
6. the kinetic modeling method considering the change of the meshing point of the bevel gear according to claim 5, wherein: in the shape function N, for an angle node, the shape function is expressed as:
7. the kinetic modeling method considering the change of the meshing point of the bevel gear according to claim 6, wherein: in the shape function N, for the intermediate node, the shape function is expressed as:
wherein, xi a 、η a 、ζ a The values of (a) are selected according to the position of each node in the iso-reference frame, and the subscript a indicates the node on the 20-node hexahedral cell.
8. The kinetic modeling method considering the change of the meshing point of the bevel gear according to claim 1, wherein: the stiffness matrix of the coupling unit of the bevel gear and the drive shaft is expressed as:
wherein K is C A diagonal matrix of 6*6, the diagonal value of which is an equally large number.
9. The kinetic modeling method considering the change of the meshing point of the bevel gear according to claim 1, wherein: the main bevel gear and the secondary bevel gear are mounted at 90 degrees in the kinetic model.
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