CN117807722A - Dynamic modeling and vibration characteristic analysis method for flexible bevel gear transmission system - Google Patents
Dynamic modeling and vibration characteristic analysis method for flexible bevel gear transmission system Download PDFInfo
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Abstract
The invention discloses a dynamic modeling and vibration characteristic analysis method of a flexible bevel gear transmission system, which 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 reduction polycondensation on the bevel gear by adopting a mode synthesis method, and modeling a transmission shaft by adopting a Timoshenko beam unit; s2, assembling a main driven bevel gear shaft system: the bevel gear and the transmission shaft are coupled through a coupling unit; s3, modeling of meshing pairs: establishing a meshing unit between two bevel gears, wherein the meshing process is simplified into a time-varying spring; s4, dynamics calculation: and calculating a dynamic response result by a Newmark-beta numerical method, setting a rotating speed interval to perform dynamic sweep frequency calculation, and drawing a vibration deformation cloud chart, a vibration speed cloud chart and a vibration acceleration cloud chart of the resonance rotating speed. The invention has accurate modeling and can obtain the vibration cloud picture at any rotating speed.
Description
Technical Field
The invention relates to the technical field of structural design of gear transmission systems, in particular to a dynamic modeling and vibration characteristic analysis method of a flexible bevel gear transmission system.
Background
In an aviation transmission system, a bevel gear is adopted to transmit non-parallel torque or speed, and in recent decades, the requirement of the aerospace field on a lightweight bevel gear is higher and higher, so that the bevel gear generally adopts a thin-wall structure in order to lighten the weight. However, the problem of web pitch diameter type vibration easily occurs in the structure, the problem of long-life operation of an engine is affected by fracture of a gear web, in the past, two modeling modes are generally available, namely, a bevel gear adopts a rigid disc to carry out dynamic modeling (rigid model), only centralized mass, centralized diameter moment of inertia and centralized polar moment of inertia are considered in the mode, accurate modeling can not be carried out on a complex bevel gear, a model which is built can not obtain a mode shape related to a bevel gear body, a great error is brought to a model building stage, and finite element software is used for building the model, but the finite element software can not carry out calculation of gear dynamics, and the dynamic analysis is carried out after a polycondensation matrix is needed to be derived by the finite element software, although the rigidity matrix and the quality matrix which are derived by adopting a finite element method can ensure that the model has high-precision mode frequency, but the structural parameters of the model are lost, the vibration cloud diagram can not be drawn, and the structure optimization direction can not be guided according to a calculation result so as to reduce vibration.
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 and vibration characteristic analysis method of a flexible bevel gear transmission system, the built dynamic model can keep the inherent properties (modal frequency and vibration mode) of the model, the modeling is accurate, the dynamic result calculated after polycondensation can be restored to the dynamic result of the whole system, the control of finite element software is eliminated, the structural properties can be kept, and the vibration cloud picture at any rotating speed can be obtained.
The dynamic modeling and vibration characteristic analysis method of the flexible bevel gear transmission system provided by the embodiment of the invention comprises the following steps of:
s1, modeling a bevel gear and a transmission shaft: modeling a bevel gear by adopting a 20-node hexahedral unit to obtain a quality matrix and a rigidity matrix of the bevel gear, performing dimension-reducing polycondensation on the bevel gear by adopting a mode synthesis method to reduce the size of a model matrix, and modeling a transmission shaft by adopting a Timoshenko beam unit;
s2, assembling a main driven 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, an engagement unit is established between two bevel gears, an engagement process is simplified into a time-varying spring, the equivalent engagement points of the two bevel gears are subjected to force interaction through the spring, the equivalent engagement points are rigidly coupled with tooth surfaces engaged at the same moment, corresponding input torque and output load are respectively applied to input and output positions of the two transmission shafts, wherein the equivalent engagement points are centers of resultant forces on the tooth surfaces, and the directions of the equivalent engagement points point to the direction of the resultant forces;
s4, dynamics calculation: calculating a dynamic response result of the bevel gear transmission system by using a Newmark-beta numerical method, setting a rotating speed interval of interest, carrying out dynamic sweep calculation on the rotating speed interval, and then drawing a vibration deformation cloud chart, a vibration speed cloud chart and a vibration acceleration cloud chart on the resonance rotating speed.
The dynamic modeling and vibration characteristic analysis method of the flexible bevel gear transmission system provided by the embodiment of the invention has at least the following beneficial effects:
the flexible modeling of the complex structure of the bevel gear is carried out based on the 20-node hexahedral unit, the dimension reduction polycondensation of the matrix is carried out by adopting a mode synthesis method, the built dynamic model can keep the inherent attribute (the mode frequency and the vibration mode) of the model, the modeling accuracy is high, the model is polycondensed by using the mode synthesis method, the calculation speed is high, the dynamic result calculated after the polycondensation can be restored to the dynamic result of the whole system, the control of finite element software is eliminated, the structural attribute can be kept, the vibration cloud picture at any rotating speed is obtained, the theoretical guidance is provided for structural optimization, and the proposed technical route can be realized by self-programming without any commercial finite element software.
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:
wherein, for an angular node, its shape function is expressed as:
for intermediate nodes, its shape function is expressed as:
in 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.
According to some embodiments of the invention, step S1 comprises the steps of:
s102, obtaining a conversion matrix T by adopting a mode synthesis method cms By means of the transformation matrix T cms The following transformations were performed:
M cms ＝T cms T MT cms ，
K cms ＝T cms T KT cms ，
thereby obtaining a mass matrix M after polycondensation of the bevel gears cms And a stiffness matrix K cms Realizing the established dimension-reducing polycondensation of the bevel gear; wherein M and K are the mass matrix and the stiffness matrix of the bevel gear as a whole.
According to some embodiments of the invention, in step S102, the conversion matrix T is obtained by using a mode synthesis method cms The method comprises the following steps:
firstly, according to the characteristics of the bevel gears, each bevel gear is provided with two interfaces, one is an engagement tooth surface of the bevel gear, the other is an inner ring surface of the bevel gear connected with the transmission shaft, and a motion equation of a substructure of the bevel gear is established under physical coordinates by 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 for which damping is not a consideration 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 motion equation 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 eigenvalues of the following free vibration equation:
by solving the eigenvalue of the free vibration equation, i natural vibration modes can be obtained, and the combination is expressed as phi by a matrix n ；
Each degree of freedom at the interface is then released in turn and the static displacement is then calculated:
from the above formula:
a i ＝-K ii -1 K ij a j ，
let a j The j elements in the matrix sequentially take unit values and the rest are 0, so as to obtain corresponding j groups of static displacement vectors, namely constraint modesState, a 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.
According to some embodiments of the invention, step S1 comprises the steps of:
s103, modeling the transmission shaft by adopting the Timoshenko beam units, dividing a plurality of Timoshenko beam units according to the structure of the transmission shaft, and performing unit assembly on each unit according to the unit numbers of the Timoshenko beam units to obtain a mass matrix and a rigidity matrix of the transmission shaft overall, wherein the mass matrix of one Timoshenko beam unit is expressed as:
a stiffness matrix of the Timoshenko beam unit is expressed as:
wherein,
wherein ρ is 2 Is the density of the transmission shaft, v is poisson ratio, E is elastic modulus, G is shear modulus of the beam unit, I s 、I p The diameter moment of inertia and the polar moment of inertia of the beam unit are respectively, A is the cross-sectional area of the beam unit, l is the length of the beam unit, m is the ratio of the inner diameter to the outer diameter of the beam unit,for simplicity, μ is a hollow correction coefficient.
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, in step S3, the stiffness matrix of the engagement 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 a load contact analysis, and V represents an engagement vector.
According to some embodiments of the invention, the meshing vector V is expressed as:
V＝[n px n py n pz 0 0 0 n gx n gy n gz 0 0 0]，
wherein subscripts p and g respectively represent a driving gear and a driven gear, n px 、n py 、n pz Is the unit component of the equivalent meshing force of the driving gear in the X, Y, Z direction, n gx 、n gy 、n gz Is a unit component of the equivalent meshing force of the driven gear in the direction X, Y, Z.
According to some embodiments of the present invention, in step S4, vibration displacement, vibration speed and vibration acceleration of the bevel gear transmission system after the dimension reduction polycondensation are calculated by using a Newmark- β numerical method, and then restored to vibration displacement, vibration speed and vibration acceleration of all degrees of freedom of the whole system, and vibration deformation cloud patterns, vibration speed cloud patterns and vibration acceleration cloud patterns are drawn according to the vibration displacement, vibration speed and vibration acceleration of the whole system.
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 view of a 20-node hexahedral unit;
FIG. 3 is a schematic view of a Timoshenko beam unit;
fig. 4 is a schematic view of a bevel gear shaft model in which (a) hexahedral unit, (b) coupling unit, (c) beam unit, (d) bearing unit;
FIG. 5 is an assembled view of a stiffness matrix;
FIG. 6 is a free mode shape of a bevel gear shaft of a finite element model, where (e) f n1,2 ＝2119Hz，(f)f n4,5 ＝2915Hz，(g)f n4,5 ＝2915Hz；
FIG. 7 is a free mode shape of the bevel gear shaft modeled herein, wherein (h) f n1,2 ＝2108Hz，(i)f n4,5 ＝2920Hz，(j)f n7,8 ＝3979Hz；
FIG. 8 is a free mode shape of a conventional rigid model bevel gear shaft, where (k) f n1,2 ＝3096Hz，(l)f n4,5 ＝5772Hz，(m)f n7,8 ＝9247Hz；
FIG. 9 is a graph of dynamic transmission error as a function of rotational speed, wherein the solid line represents a conventional rigid model and the dashed line represents the model herein;
FIG. 10 is a vibration deformation cloud of the model herein at a maximum resonance speed of 7500 rpm;
fig. 11 is a vibration deformation cloud of a conventional rigid model at a maximum resonance speed of 6900 rpm.
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.
The invention relates to a dynamic modeling and vibration characteristic analysis method of a flexible bevel gear transmission system, which adopts a 20-node hexahedral unit to establish a bevel gear and utilizes a modal synthesis method to carry out dimension-reduction polycondensation on the bevel gear so as to improve the calculation speed, adopts a Timoshenko beam unit to establish a transmission shaft, then combines the transmission shaft and the bevel gear through a coupling unit, couples a meshing tooth surface to an equivalent meshing point, finally carries out vibration characteristic analysis of the bevel gear transmission system, and draws a vibration cloud graph at a resonance rotating speed to provide theoretical guidance for the optimization design of a structure, and the method 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 built bevel gear by adopting a mode synthesis method to reduce the size of a model matrix, and simultaneously carrying out modeling on a transmission shaft by adopting a Timoshenko beam unit; the bevel gears establish a driving gear (input gear) and a driven gear (output gear), and the transmission shafts establish a driving shaft (input shaft) and a driven shaft (output shaft);
s2, assembling a main driven 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 a dynamics model, a meshing unit is established between two bevel gears (a driving gear and a driven gear), the meshing process is simplified into a time-varying spring, an equivalent meshing point of the two bevel gears is subjected to force interaction through the spring, the equivalent meshing point is rigidly coupled with tooth surfaces meshed at the same moment, corresponding input torque and 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 point is the center of resultant force on the tooth surfaces, and the direction of the resultant force is directed; modeling of the bevel gear transmission system is completed through steps S1, S2 and S3, and the built model is shown in FIG. 1.
S4, dynamics calculation: calculating a dynamic response result of the bevel gear transmission system by using a Newmark-beta numerical method, setting a rotating speed interval of interest, carrying out dynamic sweep calculation on the rotating speed interval, and then drawing a vibration deformation cloud chart, a vibration speed cloud chart and a vibration acceleration cloud chart on the resonance rotating speed, thereby providing theoretical guidance for structure optimization.
According to the dynamic modeling and vibration characteristic analysis method of the flexible bevel gear transmission system, flexible modeling of a complex structure of a bevel gear is carried out based on a 20-node hexahedral unit, dimension reduction polycondensation of a matrix is carried out by adopting a mode synthesis method, the built dynamic model can keep inherent properties (mode frequency and vibration mode) of the model, modeling accuracy is high, the model is polycondensed by using the mode synthesis method, calculation speed is high, a dynamic result calculated after polycondensation can be restored to a dynamic result of the whole system, control of finite element software is eliminated, structural properties can be kept, vibration cloud patterns at any rotating speed are obtained, theoretical guidance is provided for structural optimization, and the proposed technical route can be realized through self-programming without any commercial finite element software.
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 overall rigidity matrix of the bevel gear;
wherein, the mass matrix of a 20-node hexahedral unit is:
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.
Specifically, the shape function N is expressed as:
as shown in fig. 2, for the corner nodes of a hexahedral cell, its shape function is expressed as:
for the intermediate node of the hexahedral cell, its shape function is expressed as:
in xi a 、η a 、ζ a The values of (2) are selected according to the positions of the nodes in the reference frame, and the subscriptsa 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.
S102, obtaining a conversion matrix T by adopting 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 the bevel gears, each bevel gear is provided with two interfaces, one is the meshing tooth surface of the bevel gear, the other is the inner ring surface of the bevel gear connected with a transmission shaft, and a motion equation of a substructure of the bevel gear is established under physical coordinates by 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 eigenvalues of the following free vibration equation:
through the above solution of the eigenvalue of the free vibration equation, i natural vibration modes can be obtained, and 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 i 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.
S103, flexible modeling of the transmission shaft is conducted by adopting Timoshenko beam units, a plurality of Timoshenko beam units are divided according to the structure of the transmission shaft, and unit assembly is conducted on all the units according to unit numbers of the Timoshenko beam units so as to obtain a total mass matrix and a total 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.
In some embodiments of the invention, in step S3, in the kinetic model, an engagement unit is established between the equivalent engagement points of the two bevel gears (driving gear and driven gear), the engagement vector V being expressed as:
V＝[n px n py n pz 0 0 0 n gx n gy n gz 0 0 0]，
wherein subscripts p and g respectively represent a driving gear and a driven gear, n px 、n py 、n pz Is the unit component of the equivalent meshing force of the driving gear in the X, Y, Z direction, n gx 、n gy 、n gz Is a unit component of the equivalent meshing force of the driven gear in the X, Y, Z direction; the meshing process of the gears is simplified into a time-varying spring, and the stiffness matrix of the meshing unit can be expressed as:
K m ＝k m (t)(V) T V，
wherein k is m And (t) is the time-varying engagement stiffness, obtained by load contact analysis.
Fig. 5 is an assembled view of the model stiffness matrix herein, wherein both the driving gear (i.e., input gear) and the driven gear (i.e., output gear) have two coupling points, one is the gear inner ring node coupled to the corresponding shaft node and the other is coupled to the tooth face point, and the meshing force is applied to both of the coupled tooth face points during meshing, the total number of degrees of freedom in the proposed model system is 462, the conventional model has 318 degrees of freedom in total, while the proposed model herein has only increased 144 degrees of freedom, and the proposed model dynamics calculation does not increase much calculation time.
In some embodiments of the present invention, in step S4, vibration displacement, vibration speed and vibration acceleration of the bevel gear transmission system after the dimension reduction and polycondensation are calculated by using a Newmark- β numerical method, and then restored to vibration displacement, vibration speed and vibration acceleration of all degrees of freedom of the whole system, and according to the vibration displacement, vibration speed and vibration acceleration of the whole system, a vibration deformation cloud image, a vibration speed cloud image and a vibration acceleration cloud image can be drawn, and in addition, by setting a rotation speed interval of interest and performing dynamic sweep calculation on the rotation speed interval, a change graph of dynamic transmission errors along with rotation speed can be obtained, as shown in fig. 9, wherein the maximum amplitude of a curve represents the maximum resonance rotation speed.
Implementation case: verifying validity of models built herein
In order to verify the effectiveness of the model built herein, a modal test is performed on a bevel gear shaft with a solid body, and the modal frequencies obtained by the test are compared with those obtained by a finite element model, a model herein and a traditional rigid model, the parameter tables of a transmission shaft and a bevel gear are shown in tables 1 and 2, and table 3 is modal frequency comparison data.
The error 1 in table 3 is the error of the modal frequencies of the finite element result and the test result, the error 2 is the error of the modal frequencies of the test result and the finite element result, the error 3 is the error of the modal frequencies of the test result and the finite element result, and the error 4 is the error of the modal frequencies of the conventional rigid model and the test result, as can be seen from table 3, the modal frequencies of the test result, the finite element result and the test result are smaller, so that it can be verified that the model built in the present invention can accurately represent the inherent properties of the bevel gear shaft, while the difference between the rigid model and the test result is very large, and the inherent properties of the bevel gear shaft cannot be accurately represented, so that a larger error occurs in the modeling stage.
In addition, free mode vibration mode diagrams of the bevel gear shaft of the finite element model, the finite element model and the traditional rigid model are drawn, as shown in fig. 6, 7 and 8 respectively, and it can be seen from comparison of the diagrams that the pitch diameter vibration mode of the gear web appears for the flexible model (the finite element model and the traditional rigid model), and the rigid model is free of the pitch diameter vibration mode and only has the bending vibration mode of the shaft. It should be noted that, if the finite element method is adopted to build the model, if the dimension-reducing polycondensation is performed, structural parameters are lost, so that the mode shape cannot be drawn, and the mode shape can be obtained by using the finite element method only under the condition that the dimension-reducing polycondensation is not performed, but the calculation speed of the mode is very slow, and the dimension-reducing polycondensation is generally performed when the finite element method is adopted for modeling.
In addition, the vibration response of the bevel gear transmission system is calculated by using a Newmark-beta numerical method, dynamic sweep calculation is carried out on a rotating speed interval of 1000-15000rpm to obtain a dynamic transmission error change graph with rotating speed, which is shown in fig. 9, wherein a solid line part is a traditional rigid model, a dotted line represents a text model, a curve maximum amplitude represents a maximum resonance rotating speed, vibration deformation cloud graphs of the text model at the maximum resonance rotating speed of 7500rpm and a rigid model at the maximum resonance rotating speed of 6900rpm are respectively drawn, as shown in fig. 10 and 11, the pitch diameter vibration mode of the bevel gear is easier to cause resonance for a bevel gear with a web, the pitch diameter vibration mode of the text provides that the maximum amplitude of the model in the range of 1000-15000rpm appears as the pitch diameter vibration mode is shown, and the traditional rigid model is too simplified for the bevel gear to cause bending mode of a gear shaft during modeling, so that the model established in the text can be verified more accurately compared with the traditional rigid model.
TABLE 1 parameters of each shaft segment of the drive shaft
In Table 1, the column of lengths shows that a drive shaft is divided axially into shaft sections 9mm, 51mm, 134mm, 15mm, 35mm long.
TABLE 2 Master-slave bevel gear parameters
Table 3 modal frequency contrast
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 (10)
1. The dynamic modeling and vibration characteristic analysis method of the flexible bevel gear transmission system 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 to obtain a quality matrix and a rigidity matrix of the bevel gear, performing dimension-reducing polycondensation on the bevel gear by adopting a mode synthesis method to reduce the size of a model matrix, and modeling a transmission shaft by adopting a Timoshenko beam unit;
s2, assembling a main driven 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, an engagement unit is established between two bevel gears, an engagement process is simplified into a time-varying spring, the equivalent engagement points of the two bevel gears are subjected to force interaction through the spring, the equivalent engagement points are rigidly coupled with tooth surfaces engaged at the same moment, corresponding input torque and output load are respectively applied to input and output positions of the two transmission shafts, wherein the equivalent engagement points are centers of resultant forces on the tooth surfaces, and the directions of the equivalent engagement points point to the direction of the resultant forces;
s4, dynamics calculation: calculating a dynamic response result of the bevel gear transmission system by using a Newmark-beta numerical method, setting a rotating speed interval of interest, carrying out dynamic sweep calculation on the rotating speed interval, and then drawing a vibration deformation cloud chart, a vibration speed cloud chart and a vibration acceleration cloud chart on the resonance rotating speed.
2. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method 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 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.
3. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method according to claim 2, wherein: the shape function N is expressed as:
wherein, for an angular node, its shape function is expressed as:
for intermediate nodes, its shape function is expressed as:
in 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.
4. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method according to claim 2, wherein: step S1 comprises the steps of:
s102, obtaining a conversion matrix T by adopting a mode synthesis method cms By means of the transformation matrix T cms The following transformations were performed:
M cms ＝T cms T MT cms ，
K cms ＝T cms T KT cms ，
thereby obtaining a mass matrix M after polycondensation of the bevel gears cms And a stiffness matrix K cms Realizing the established dimension-reducing polycondensation of the bevel gear; wherein M and K are the mass matrix and the stiffness matrix of the bevel gear as a whole.
5. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method according to claim 4, wherein: in step S102, the conversion matrix T is obtained by using a mode synthesis method cms The method comprises the following steps:
firstly, according to the characteristics of the bevel gears, each bevel gear is provided with two interfaces, one is an engagement tooth surface of the bevel gear, the other is an inner ring surface of the bevel gear connected with the transmission shaft, and a motion equation of a substructure of the bevel gear is established under physical coordinates by 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 for which damping is not a consideration 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 motion equation 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 eigenvalues of the following free vibration equation:
by solving the eigenvalue of the free vibration equation, i can be obtainedNatural vibration mode, combined into matrix expressed as phi n ；
Each degree of freedom at the interface is then released in turn and the static displacement is then 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.
6. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method according to claim 1, wherein: step S1 comprises the steps of:
s103, modeling the transmission shaft by adopting the Timoshenko beam units, dividing a plurality of Timoshenko beam units according to the structure of the transmission shaft, and performing unit assembly on each unit according to the unit numbers of the Timoshenko beam units to obtain a mass matrix and a rigidity matrix of the transmission shaft overall, wherein the mass matrix of one Timoshenko beam unit is expressed as:
a stiffness matrix of the Timoshenko beam unit is expressed as:
wherein,
wherein ρ is 2 Is the density of the transmission shaft, v is poisson ratio, E is elastic modulus, G is shear modulus of the beam unit, I s 、I p The diameter moment of inertia and the polar moment of inertia of the beam unit are respectively, A is the cross-sectional area of the beam unit, l is the length of the beam unit, m is the ratio of the inner diameter to the outer diameter of the beam unit,for simplicity, μ is a hollow correction coefficient.
7. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method according to claim 1, wherein: the stiffness matrix of the coupling unit of the bevel gear and the transmission shaft is expressed as:
wherein K is C A diagonal matrix of 6*6, the diagonal value of which is an equally large number.
8. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method according to claim 1, wherein: in step S3, the stiffness matrix of the engagement 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 a load contact analysis, and V represents an engagement vector.
9. The flexible bevel gear transmission system dynamics modeling and vibration characteristics analysis method according to claim 8, wherein: the meshing vector V is expressed as:
V＝[n px n py n pz 0 0 0 n gx n gy n gz 0 0 0]，
wherein subscripts p and g respectively represent a driving gear and a driven gear, n px 、n py 、n pz Is the unit component of the equivalent meshing force of the driving gear in the X, Y, Z direction, n gx 、n gy 、n gz Is a unit component of the equivalent meshing force of the driven gear in the direction X, Y, Z.
10. The flexible bevel gear transmission system dynamics modeling and vibration characteristic analysis method according to claim 1, wherein: in the step S4, vibration displacement, vibration speed and vibration acceleration of the bevel gear transmission system after dimension reduction and polycondensation are calculated through a Newmark-beta numerical method, and then restored to vibration displacement, vibration speed and vibration acceleration of all degrees of freedom of the whole system, and vibration deformation cloud pictures, vibration speed cloud pictures and vibration acceleration cloud pictures are drawn according to the vibration displacement, vibration speed and vibration acceleration of the whole system.
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