CN117807723A - Method for rapidly deviating resonance rotating speed of flexible bevel gear transmission system - Google Patents
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Abstract
The invention discloses a method for rapidly deviating the resonance rotating speed of a flexible bevel gear transmission system, which comprises the following steps: s1, modeling a bevel gear transmission system; s2, dangerous mode calculation: calculating the modal frequency of the bevel gear transmission system, if the modal orders exist in the range of 15% before and after the working rotation speed, calculating the modal strain energy of each order of modes in the range of 15% before and after the working rotation speed, comparing the modal strain energy with the total modal strain energy, analyzing whether the modal energy is dangerous modal frequency according to the duty ratio result, and drawing the vibration mode of the dangerous modal frequency; s3, structural parameter optimization: and carrying out parametric study on structural parameters, bearing positions and rigidity of the bevel gear and the transmission shaft, so that dangerous mode frequency is far away from the working rotating speed. The invention can rapidly identify the dangerous mode frequency, draw the mode shape of the system, and rapidly deviate the resonance rotating speed from the working rotating speed by modifying the structural parameters, the bearing position and the rigidity of the bevel gear and the transmission shaft when the dangerous mode frequency exists near the working rotating speed.
Description
Technical Field
The invention relates to the technical field of structural design of gear transmission systems, in particular to a method for rapidly deviating the resonance rotating speed 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, so that the long-life operation of an engine is affected by fracture of a gear web, in the past study, two modeling modes are generally available, namely, a bevel gear adopts a rigid disc to perform dynamic modeling (rigid model), the mode cannot be accurately modeled on a complex bevel gear, the mode vibration mode related to a bevel gear body cannot be obtained, a great error is brought to a model building stage, and finite element software is adopted to build the model, but the finite element software cannot perform calculation of gear dynamics, programming is required to be performed after a polycondensation matrix is derived by the finite element software, dynamic analysis is required, although the rigidity matrix and the quality matrix derived by the finite element method can ensure that the model has high-precision mode frequency, the mode vibration mode cannot be calculated by losing structural parameters of the model, and the obtained mode frequency cannot judge which frequencies are easy to excite resonance, namely, dangerous mode frequency easy to excite resonance cannot be identified, so that the design cannot be guided to deviate the resonant rotation speed.
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 method for rapidly deviating the resonance rotating speed of a flexible bevel gear transmission system, which can rapidly identify dangerous mode frequencies which are easy to excite resonance, and can draw the mode shape of the system, and when the dangerous mode frequencies exist near the working rotating speed, the resonance rotating speed is rapidly deviated from the working rotating speed by modifying the structural parameters, the bearing positions and the rigidity of a bevel gear and a transmission shaft.
The method for rapidly deviating the resonance rotating speed of the flexible bevel gear transmission system comprises the following steps of:
s1, modeling a bevel gear transmission system, which comprises the following steps:
s100, 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 model matrix, and modeling a transmission shaft by adopting a Timoshenko beam unit;
s110, 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;
s120, modeling of meshing pairs: in the dynamics model, a meshing unit is established between the two bevel gears, and the meshing process is simplified into a spring;
s2, dangerous mode calculation: calculating the modal frequency of each step of the bevel gear transmission system, if the modal order exists in the range of 15% of the front and rear working speeds, calculating the modal strain energy of each step in the range of 15% of the front and rear working speeds and comparing the modal strain energy with the total modal strain energy to obtain the modal strain energy duty ratio of each step, analyzing whether the modal frequency of the corresponding step is a dangerous modal frequency which is easy to excite resonance according to the modal strain energy duty ratio result, and drawing the vibration mode of the dangerous modal frequency;
s3, structural parameter optimization: and carrying out parameterization research on structural parameters, bearing positions and rigidity of the bevel gear and the transmission shaft according to the vibration mode of the dangerous mode frequency, so that the dangerous mode frequency is far away from the working rotating speed or is converted into the mode frequency which is not easy to excite resonance.
The method for rapidly deviating the resonance rotating speed of the flexible bevel gear transmission system has at least the following beneficial effects:
the flexible modeling of the bevel gear complex structure is carried out based on the 20-node hexahedron unit, the modeling of the transmission shaft is carried out based on the Timoshenko beam unit, the modeling is accurate, and the inherent properties (modal frequency and vibration mode) of the model can be reserved, so that the system modal vibration mode can be drawn, guidance is provided for structure optimization, the model is condensed by adopting a modal synthesis method after the bevel gear is modeled, the calculation speed is high, and the time of the whole process can be greatly reduced; by calculating the modal frequencies of each order of the bevel gear transmission system, analyzing whether modal orders exist in the range of 15% before and after the working rotation speed, and calculating the modal strain energy duty ratio of each order of modes in the range of 15% before and after the working rotation speed, the dangerous modal frequency which is easy to excite resonance can be rapidly identified according to the modal strain energy duty ratio result, then the vibration mode under the dangerous modal frequency is drawn, namely, what is the part which is easy to cause resonance according to the vibration mode analysis, and then the structural parameters of the part which is easy to cause resonance are optimized, so that the dangerous modal frequency is far away from the working rotation speed or the dangerous modal frequency is converted into the modal frequency which is difficult to excite resonance, and the purpose of deviating the resonance rotation speed from the working rotation speed can be achieved.
According to some embodiments of the invention, in step S2, the modal strain energy in the j-th order mode is expressed as:
in phi mj Is the feature vector related to engagement in the j th order feature vector, K m Is a stiffness matrix of the engagement unit.
According to some embodiments of the invention, in step S2, the total modal strain energy is expressed as:
wherein K is z Is the overall rigidity matrix phi of the bevel gear transmission system j Representing the feature vector of the j-th order.
According to some embodiments of the invention, step S100 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 S100 comprises the steps of:
s102, adoptObtaining a conversion matrix T by 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 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 ：
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 operations are performedObtaining 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, in step S110, the stiffness matrix of the coupling unit 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 S120, the stiffness matrix of the engagement unit is expressed as:
K m ＝k m (V) T V，
wherein k is m As an average value of time-varying engagement stiffness, obtained by load contact analysis, 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.
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 schematic view of a 20-node hexahedral unit;
FIG. 2 is a schematic view of a Timoshenko beam unit;
FIG. 3 is a schematic view of a bevel gear shaft model in which (a) hexahedral units, (b) coupling units, (c) beam units, (d) bearing units;
FIG. 4 is an assembly diagram of a system stiffness matrix;
FIG. 5 is a top view of the primary pattern model herein;
FIG. 6 is a side view of the primary pattern model herein;
FIG. 7 is modal strain energy duty cycle for each order of the primary scheme model herein;
FIG. 8 is the mode shape of the 30 th order of the original pattern model herein;
FIG. 9 is a top view of the novel scheme model herein;
FIG. 10 is a side view of the novel solution model herein;
FIG. 11 is modal strain energy duty cycle for each order of the novel scheme model herein;
fig. 12 is the mode shape of the 30 th order of the novel scheme model herein.
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 method for rapidly deviating the resonance rotating speed of a flexible bevel gear transmission system, which comprises the following steps:
s1, modeling a bevel gear transmission system, wherein the step S1 specifically comprises the following steps:
s100, 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);
s110, 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, and applying rigidity of a bearing on a bearing node on the transmission shaft to simulate the bearing, wherein obviously, a driving gear is coupled with a driving shaft, and a driven gear is coupled with a driven shaft;
s120, modeling of meshing pairs: in the dynamics model, a meshing unit is established between two bevel gears (a driving gear and a driven gear), the meshing process is simplified into springs, the equivalent meshing points of the two bevel gears are subjected to force interaction through the 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 (a driving shaft and a driven shaft) of the two transmission shafts; modeling of the bevel gear transmission system is carried out through steps S100, S110 and S120;
s2, dangerous mode calculation: calculating the modal frequency of each order of the bevel gear transmission system, if the modal order exists within 15% of the range before and after the working rotation speed (namely, the modal frequency of a certain order or multiple orders of modes exists within 15% of the range before and after the frequency corresponding to the working rotation speed), calculating the modal strain energy of each order of modes within 15% of the range before and after the working rotation speed, comparing the modal strain energy with the total modal strain energy to obtain the modal strain energy duty ratio of each order, analyzing whether the modal frequency of the corresponding order is the dangerous modal frequency which is easy to excite resonance according to the modal strain energy duty ratio result, and drawing the vibration mode of the dangerous modal frequency; the rotation speed in the range of 15% before and after the working rotation speed is easy to cause resonance, the dangerous mode frequency is the frequency easy to cause resonance, the rotation speed corresponding to the dangerous mode frequency is the resonance rotation speed, and the mode frequency of the order with high mode strain energy ratio is easy to cause resonance;
s3, structural parameter optimization: carrying out parameterization research on structural parameters, bearing positions, rigidity and the like of the bevel gear and the transmission shaft according to the vibration mode of the dangerous mode frequency, so that the dangerous mode frequency is far away from the working rotating speed or the dangerous mode frequency is converted into the mode frequency which is not easy to excite resonance, thereby realizing the purpose of deviating the resonance rotating speed from the working rotating speed; the parametric study on the structural parameters, bearing positions, rigidity and the like of the bevel gear and the transmission shaft according to the vibration mode of the dangerous mode frequency refers to confirming what is the part causing resonance according to the vibration mode analysis of the dangerous mode frequency, and then modifying the structural parameters of the part causing resonance to perform structural optimization.
According to the flexible bevel gear transmission system resonance rotating speed rapid deviation method, flexible modeling of a complex structure of a bevel gear is performed based on a 20-node hexahedral unit, modeling of a transmission shaft is performed based on a Timoshenko beam unit, modeling is accurate, inherent properties (modal frequency and vibration mode) of a model can be reserved, accordingly, the system modal mode can be drawn, guidance is provided for structure optimization, and the model is subjected to polycondensation by adopting a modal synthesis method after the bevel gear is modeled, so that calculation speed is high, and time of the whole flow can be greatly reduced; and by calculating the modal frequency of each order of the bevel gear transmission system, analyzing whether the modal order exists in the range of 15% before and after the working rotation speed, and calculating the modal strain energy duty ratio of each order of modes in the range of 15% before and after the working rotation speed, the dangerous modal frequency which is easy to excite resonance can be rapidly identified according to the modal strain energy duty ratio result, and then the vibration mode under the dangerous modal frequency is drawn, namely, according to what the vibration mode analysis causes the resonance part, the structural parameters of the resonance-causing part are optimized, so that the dangerous modal frequency is far away from the working rotation speed or the dangerous modal frequency is converted into the modal frequency which is difficult to excite resonance, and the purpose of deviating the resonance rotation speed from the working rotation speed can be realized.
It will be appreciated that in some particular embodiments, in step S2, the modal strain energy U in the jth order mode mj Expressed as:
in phi mj Is the feature vector related to engagement in the j th order feature vector, K m Is a stiffness matrix of the engagement unit;
total modal strain energy U T Expressed as:
in phi j Representing feature vectors of the j th order, K z Is the overall rigidity matrix of the bevel gear transmission system, the overall rigidity matrix can be obtained by assembling all parts of the bevel gear transmission system, and fig. 4 is a rigidity matrix assembly diagram of the bevel gear transmission system model established in the text;
the modal strain energy duty ratio of the j th order is expressed as U mj /U T 。
In some specific embodiments of the present invention, step S100 includes 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 a Jacobian matrix, and ζ, η and ζ are integral point coordinates;
wherein the shape function N is expressed as:
as shown in fig. 1, 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 (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.
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:
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 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. 2, 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 some specific embodiments of the present invention, referring to fig. 3, in step S110, the bevel gear and the drive shaft are coupled by the coupling unit, and the coupling node of the bevel gear inner ring 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 specific embodiments of the present invention, in step S120, in the kinetic model, an engagement unit is established between the equivalent engagement points of the two bevel gears (driving gear and driven gear), and since the equivalent engagement points have considered the torque caused by the gear engagement, the engagement vector V involves only translational degrees of freedom, and 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 X, Y, Z direction; the meshing process of the gears is simplified to springs, and the stiffness matrix of the meshing units can be expressed as:
K m ＝k m (V) T V，
wherein k is m Is an average of time-varying engagement stiffness obtained by load contact analysis.
Implementation case:
the method is adopted to build a model of a traditional bevel gear system, wherein a transmission shaft is firstly built into a solid shaft, a bevel gear parameter table is shown in table 1 (in order to distinguish from a follow-up scheme, a scheme of building the transmission shaft into the solid shaft is called an original scheme), the original scheme model is shown in fig. 5 and 6, the mode frequencies of each order mode of the original scheme model are calculated, the mode strain energy duty ratio of 0-50 order modes of the original scheme model is calculated for more visual data, and a bar chart of the mode strain energy duty ratio of each order is drawn and is shown in fig. 7.
TABLE 1 bevel gear parameters
When the input operation speed is 6700rpm, the mode frequency which can excite resonance near the rotation speed needs to deviate to enable the system to stably operate, when the number of teeth of the driving gear is 55, the mode frequency of the system corresponding to 6700rpm is 6700/60×55=6142 Hz, therefore, the mode frequency which is 15% before and after 6142Hz needs to deviate, and according to fig. 7, the mode strain energy ratio of the 30 th order is high, and the mode frequency of the corresponding system is in the range of 15% before and after 6142Hz, so the mode frequency of the 30 th order is dangerous mode frequency, the mode frequency of the 30 th order needs to deviate to meet the condition of stable operation, the vibration mode of the 30 th order of the original scheme is drawn, as shown in fig. 8, the left side of the drawing is the driven gear and the shaft, the right side of the drawing is the joint diameter vibration mode of the driving gear and the driven shaft, and the bending coupling vibration mode of the driven shaft can be seen from the drawing.
The original solid shaft is changed into a hollow shaft by adjusting the hollow degree of the shaft, the position of the bearing is adjusted, the model of the bearing is changed, namely, the rigidity of the bearing is changed, a new scheme is obtained, parameters of the transmission shaft after adjustment are shown in table 2 (in table 2, the column of the length represents that one transmission shaft is divided into a plurality of shaft sections with the lengths of 9mm, 51mm, 134mm, 15mm and 35mm along the axial direction), and a new scheme model is shown in fig. 9 and 10. As shown in FIG. 11, the strain energy of each order mode of the novel scheme model is very low, the meshing strain energy of the 30 th order mode does not belong to the dangerous mode, the frequency of the adjacent first order dangerous mode is 4609Hz, the frequency of the adjacent first order dangerous mode is 7094Hz, the frequency of the adjacent first order dangerous mode is more than 15% of the frequency of 6142Hz, and the requirement that the working frequency avoids the resonance frequency range in engineering is met. The vibration mode of the 30 th order of the new scheme is drawn as shown in fig. 12, the vibration mode is changed into the bending mode of the driven shaft and the pitch diameter vibration mode of the driven bevel gear, and the current vibration mode is not easy to excite resonance.
Table 2 parameters of each shaft section of the drive shaft
In addition, for the new scheme, the model is built by using the method and the model is built by using the finite element method and is compared with the traditional rigid model, the modal frequencies of the first 20 th order system are shown in table 3, wherein the error 1 is the error of the text model and the finite element model, the error 2 is the error of the text model and the rigid model, the error 1 of the first 20 th order is less than 2%, and the error of the first 11 th order of the rigid model is relatively smaller due to the bending frequency of the shaft, but the error of the 12 th order to the 20 th order is very large, namely the rigid model cannot correctly reflect the bevel gear transmission system model.
In addition, it can be seen from table 3 that there are 20-order modal frequencies at 3418Hz, and the interval between adjacent modal frequencies is very small, so that the resonant rotational speed cannot be far away from the working rotational speed by directly using the modal frequencies of each order to adjust the modal frequency in the design stage, because the modal orders about 15% of the working rotational speed may cause resonance, but if it is not known which one is caused, all the modal frequencies need to be moved, if the interval is very small, the modal frequencies in the area cannot be moved, and the method can rapidly identify the dangerous modal frequency by calculating the modal strain energy ratio in each order of modes within 15% of the range before and after the working rotational speed, and then drawing the vibration mode of the dangerous modal frequency for parametric study.
Table 3 comparison of modal frequencies of the system
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. A method for rapidly deviating the resonance rotating speed of a flexible bevel gear transmission system is characterized by comprising the following steps:
s1, modeling a bevel gear transmission system, which comprises the following steps:
s100, 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 model matrix, and modeling a transmission shaft by adopting a Timoshenko beam unit;
s110, 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;
s120, modeling of meshing pairs: in the dynamics model, a meshing unit is established between the two bevel gears, and the meshing process is simplified into a spring;
s2, dangerous mode calculation: calculating the modal frequency of each step of the bevel gear transmission system, if the modal order exists in the range of 15% of the front and rear working speeds, calculating the modal strain energy of each step in the range of 15% of the front and rear working speeds and comparing the modal strain energy with the total modal strain energy to obtain the modal strain energy duty ratio of each step, analyzing whether the modal frequency of the corresponding step is a dangerous modal frequency which is easy to excite resonance according to the modal strain energy duty ratio result, and drawing the vibration mode of the dangerous modal frequency;
s3, structural parameter optimization: and carrying out parameterization research on structural parameters, bearing positions and rigidity of the bevel gear and the transmission shaft according to the vibration mode of the dangerous mode frequency, so that the dangerous mode frequency is far away from the working rotating speed or is converted into the mode frequency which is not easy to excite resonance.
2. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S2, the modal strain energy in the j-th order mode is expressed as:
in phi mj Is the feature vector related to engagement in the j th order feature vector, K m Is the stiffness moment of the engagement unitAn array.
3. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S2, the total modal strain energy is expressed as:
wherein K is z Is the overall rigidity matrix phi of the bevel gear transmission system j Representing the feature vector of the j-th order.
4. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: step S100 includes 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 momentAnd the matrix, xi, eta and zeta are integral point coordinates.
5. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 4, 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.
6. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 4, wherein: step S100 includes 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 bevel gearsAn overall mass matrix and a stiffness matrix.
7. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 6, 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 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 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.
8. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S110, the stiffness matrix of the coupling unit 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 method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S120, the stiffness matrix of the engagement unit is expressed as:
K m ＝k m (V) T V，
wherein k is m As an average value of time-varying engagement stiffness, obtained by load contact analysis, V represents an engagement vector.
10. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 9, 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.
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