CN110990967B - Method and system for setting working rotating speed of gear transmission system - Google Patents
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Abstract
The invention discloses a method and a system for setting the working rotating speed of a gear transmission system. Because the gear transmission model has higher precision compared with the existing gear concentrated quality model, and the natural frequency precision of the gear transmission system solved by the gear transmission model is higher, the precision of the natural frequency solved according to the model with higher precision is more accurate, the resonance of the gear can be avoided by setting the working rotating speed of the gear according to the natural frequency, the safety of the gear working is improved, and the technical problem that the actual working frequency is not avoided by the existing working rotating speed set according to the natural frequency with low precision, so that the gear generates resonance is solved.
Description
Technical Field
The invention belongs to the field of gear intrinsic parameter acquisition, and particularly relates to a method and a system for setting the working rotating speed of a gear transmission system.
Background
Gear transmissions are widely used in a variety of mechanical devices. The dynamic performance of the system has great influence on the whole machine equipment system. The inherent characteristic is one of basic dynamic characteristics of the gear system, the inherent characteristic has guiding significance on the dynamic response of the system, and meanwhile, the inherent characteristic analysis is also beneficial to guiding the early design work of the gear system structure, so that resonance is avoided. At present, for the dynamic modeling analysis of the gear transmission system, a finite element method is most commonly used. I.e. the gear shaft is built by using the beam unit, and the gear is built by using a concentrated mass method. The connection is made by a gear engagement unit and a bearing unit. And finally, solving subsequent inherent frequency and dynamic response by assembling to obtain a mass matrix, a damping matrix and a rigidity matrix of the whole system. Because the gear is simplified into concentrated mass, the coupling of a transmission shaft and a gear vibration mode is not considered in the model, when the gear shaft is short or the gear diameter is large, the precision of the whole gear shaft finite element model is reduced and even distorted, so that the natural frequency obtained by the gear transmission system and the actual natural frequency of the gear have large errors, the natural frequency is used as the basis for setting the working rotating speed (in order to avoid the damage to the gear caused by the resonance of the gear due to the consistency of the working rotating speed and the natural frequency, and the working rotating speed is set to be avoided to be consistent with the natural frequency), once the obtained natural frequency and the actual natural frequency have large errors, the working rotating speed set according to the obtained natural frequency does not avoid the actual natural frequency and is set to be consistent with the actual natural frequency, the gear is inevitably resonated to cause the working damage of the gear, and the service life of the gear is shortened.
Disclosure of Invention
According to the method and the system for setting the working rotating speed of the gear transmission system, the superunit model of the gear wheel body is established, the superunit model, the established shaft unit model, the established bearing model and the established gear meshing model are used for establishing the gear transmission model with higher precision, the natural frequency of the gear transmission system is solved through the gear transmission model, and the working rotating speed of the gear transmission system is set according to the natural frequency. Because the gear transmission model is higher than the precision of the existing centralized mass model, the precision of the natural frequency solved according to the model with higher precision is more accurate, and the working rotating speed of the gear is set according to the natural frequency, so that the resonance of the gear can be avoided, and the technical problems that the actual working frequency is not avoided by the existing working rotating speed set according to the natural frequency with low precision, the gear generates resonance, and the service life of the gear is shortened are solved.
In order to solve the technical problems, the technical scheme provided by the invention is as follows:
a method for setting the operating speed of a gear transmission system comprises the following steps:
obtaining a shaft unit parameter, a bearing parameter and an engagement parameter of the gear in historical data, and respectively constructing a shaft unit model, a bearing model and a gear engagement model according to the shaft unit parameter, the bearing parameter and the engagement parameter; establishing a super-unit model of the gear wheel body according to the shaft unit model, the bearing model and the gear meshing model;
assembling the shaft unit model, the bearing model, the gear meshing model and the super unit model according to a gear transmission structure to obtain a gear transmission model;
inputting the shaft unit parameters, the bearing parameters and the gear meshing parameters of the gear transmission system to be set into the traditional gear model, solving the natural frequency of the gear transmission system, and calculating and setting the working rotating speed of the gear transmission system according to the natural frequency.
Preferably, the axle unit model is:
wherein M is S ，C s ，K S And F s Respectively a mass matrix, a damping matrix, a rigidity matrix and a load matrix of the shaft unit;
and q is S The acceleration matrix, the speed matrix and the displacement matrix of the shaft unit are respectively as follows:
wherein x, y and z are translation freedom degrees of the axis unit in a first direction, a second direction and a third direction of the system respectively, the first direction, the second direction and the third direction are mutually vertical, i is the ith node of the axis unit, theta represents the rotation freedom degree of the axis unit,
is an angular acceleration of the shaft unit,
is the angular velocity of the shaft unit.
Preferably, the bearing model is:
wherein, K b For the stiffness matrix of the bearing, subscript b denotes the bearing, k ii ，
Respectively, the displacement stiffness coefficient and the angular stiffness coefficient, theta, of the support bearing i For the rotational degrees of freedom in the translational direction of the system i, i e (x, y, z), the indices x, y and z represent the translational degrees of freedom of the bearing in the first, second and third orientations of the system, respectively.
Preferably, the gear engagement model is:
wherein, F m Is the meshing force between gear pairs, k m For time-varying meshing stiffness, gamma m0 ，γ m1 Are all piecewise nonlinear clearance indicating functions, determined by the relative displacement in the direction of the meshing line and the backlash, gamma m0 For determining elastic and damping forces, gamma, between gear pairs under the influence of play m1 The impact force between the gear pairs under the action of the clearance is determined; delta. For the preparation of a coating m Relative displacement in the direction of the meshing line; c. C m Is engaged damping; b is a clearance at the side of the half tooth,
is the meshing line relative velocity; m is a group of m ，C m ，K m Respectively a mass matrix, a meshing damping matrix and a meshing rigidity matrix;
q m acceleration, speed and displacement vector matrixes of the gear pair are respectively; e (t) is a static transfer error displacement function,
is a static transfer error velocity function; f. of m A coordinate transformation matrix is adopted;
f m ＝(c b s,cc b ,-s b ,r p ss b ,cr p s b ,c b r p ,-c b s,-cc b ,s b ,r g ss b ,cr g s b ,c b r g ) T ；
wherein, c b Is the cosine of the helix angle, s is the sine of the helix angle, c is the cosine of the end face pressure angle, s b The sine value of the end face pressure angle is shown, beta is a helical angle, phi is the end face pressure angle, and r represents the radius of a base circle; r is p ,r g The base radius of the driving and driven gears are respectively.
c b ＝cosβ,s b ＝sinβ,c＝cosφ,s＝sinφ；
Wherein q is m ＝(x p ,y p ,z p ,θ px ,θ py ,θ pz ,x g ,y g ,z g ,θ gx ,θ gy ,θ gz ) T X, y and z respectively represent the translational freedom degree of the system, and theta is the rotational freedom degree; subscripts p, g represent drive and driven gears, respectively, and an upper target tau represents a transposition factor;
wherein the content of the first and second substances,
wherein, the first and the second end of the pipe are connected with each other,
preferably, the building of the superunit model of the gear wheel body is completed by adopting a dynamic substructure method, which comprises the following steps:
establishing a coupling node at the circle center of an inner ring of the gear, rigidly constraining the coupling node and the inner ring of the gear, setting the coupling node as a main degree of freedom, and obtaining an equivalent mass matrix and a stiffness matrix of the gear by a dynamic substructure method, wherein the equivalent mass matrix and the stiffness matrix of the gear are super-unit models.
Preferably, the shaft unit model, the bearing model, the gear meshing model and the superunit model are assembled according to a gear transmission structure to obtain the gear transmission model; the method specifically comprises the following steps:
assembling the shaft unit model, the bearing model, the gear meshing model and the mass matrix and the rigidity matrix of the super unit model to obtain the mass matrix and the rigidity matrix of the whole gear transmission system; namely:
wherein M, C, K respectively represent the mass matrix, damping matrix, rigidity matrix of the whole system, f (q) represents the nonlinear displacement function caused by the clearance, and the function gamma is indicated by the nonlinear clearance m0 It is decided that F (t) represents the external torque excitation and the internal excitation.
Preferably, solving the natural frequency of the gear transmission system with the natural frequency to be obtained is realized by a matrix eigenvalue function, and the matrix eigenvalue function specifically includes:
[V,D]＝eig(K,M)；
wherein V is the solved eigenvector, D is the solved eigenvalue, K is the overall stiffness matrix, and M is the overall mass matrix.
A computer system comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of any of the methods described above when executing the computer program.
The invention has the following beneficial effects:
1. according to the method and the system for setting the working rotating speed of the gear transmission system, the gear transmission model with higher precision is built by building the superunit model of the gear wheel body, the built shaft unit model, the built bearing model and the built gear meshing model, the natural frequency of the gear transmission system is solved through the gear transmission model, and the working rotating speed of the gear transmission system is set according to the natural frequency. Because the gear transmission model is higher than the precision of the existing centralized mass model, the precision of the natural frequency solved according to the model with higher precision is more accurate, and the working rotating speed of the gear is set according to the natural frequency, so that the resonance of the gear can be avoided, and the technical problems that the actual working frequency is not avoided by the existing working rotating speed set according to the natural frequency with low precision, the gear generates resonance, and the service life of the gear is shortened are solved.
In addition to the objects, features and advantages described above, other objects, features and advantages of the present invention are also provided. The present invention will be described in further detail below with reference to the accompanying drawings.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of a method of setting an operating speed of a gear system in accordance with the present invention;
FIG. 2 is a schematic view of an axis unit model in a preferred embodiment of the present invention;
FIG. 3 is a schematic view of a gear mesh model in a preferred embodiment of the present invention;
FIG. 4 is a schematic illustration of gear superunit modeling in a preferred embodiment of the present invention;
FIG. 5 is a schematic assembly view of the gear system in the preferred embodiment of the present invention;
fig. 6 is a schematic view of a three-dimensional model of a spur gear in a preferred embodiment of the present invention.
Detailed Description
Embodiments of the invention will be described in detail below with reference to the drawings, but the invention can be implemented in many different ways as defined and covered by the claims.
The first embodiment is as follows:
as shown in FIG. 1, the invention discloses a method for setting the working rotating speed of a gear transmission system, which comprises the following steps:
obtaining shaft unit parameters, bearing parameters and meshing parameters of the gear in historical data, and respectively constructing a shaft unit model, a bearing model and a gear meshing model according to the shaft unit parameters, the bearing parameters and the meshing parameters; establishing a superunit model of the gear wheel body according to the shaft unit model, the bearing model and the gear meshing model;
assembling the shaft unit model, the bearing model, the gear meshing model and the super unit model according to a gear transmission structure to obtain a gear transmission model;
inputting the shaft unit parameters, the bearing parameters and the gear meshing parameters of the gear transmission system to be set into the traditional gear model, solving the natural frequency of the gear transmission system, and calculating and setting the working rotating speed of the gear transmission system according to the natural frequency.
In addition, in this embodiment, a computer system is also disclosed, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, and when the processor executes the computer program, the steps of the method in any one of the above embodiments are implemented.
According to the method and the system for setting the working rotating speed of the gear transmission system, the dynamic substructure method is adopted to establish the superunit model of the gear wheel body, the gear transmission model with higher precision is established through the superunit model of the gear wheel body, the natural frequency of the gear transmission system is solved through the gear transmission model, and the working rotating speed of the gear is set according to the natural frequency. Because the gear drive model is higher than the precision of the current concentrated quality model, consequently, the precision of the natural frequency of solving according to the higher model of precision is more accurate, according to the resonance's of natural frequency resonance's emergence can more be avoided to the operating speed that the natural frequency set up the gear, improves the security of gear work, and then solves the operating speed that current natural frequency that sets up according to the precision is not high and does not avoid actual operating frequency, causes the gear to produce resonance for the technical problem of the shortening of gear life.
Example two:
the second embodiment is the preferred embodiment of the first embodiment, and is different from the first embodiment in that how to construct the shaft unit model, the bearing model, the gear meshing model, and the superunit model, how to assemble the shaft unit model, the bearing model, the gear meshing model, and the superunit model according to the gear transmission structure to obtain the gear transmission model, and how to solve the gear transmission model are refined, and the method specifically includes the following steps:
wherein constructing the axle unit model comprises the steps of:
as shown in fig. 2, a finite element model of a transmission shaft is constructed by using Timoshenko (iron-wood-sinco) beam elements, in which one beam element has two nodes, each having 6 degrees of freedom, corresponding to three translational degrees of freedom and three rotational degrees of freedom, respectively. Therefore, the axis unit model (i.e., the differential equation of motion of the axis unit) is constructed as:
wherein the superscript s denotes the axis unit, M s ，C s ，K s And F s Respectively a mass matrix, a damping matrix, a rigidity matrix and a load matrix of the shaft unit;
and q is s Acceleration matrix, velocity matrix and position of the axle unit respectivelyThe matrix shifting specifically comprises:
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) )；
wherein x, y and z are translation freedom degrees of the axis unit in a first direction, a second direction and a third direction of the system respectively, i is the ith node of the axis unit, theta represents the rotation freedom degree of the axis unit,
is an angular acceleration of the shaft unit,
is the angular velocity of the shaft unit.
Wherein constructing the bearing model comprises:
since the mass of the bearing is relatively small, the mass of the bearing is generally ignored during processing, and only the support stiffness is considered. In gerotor systems, rolling bearings versus rolling bearings are typically used, and therefore can be modeled with a 6x6 stiffness matrix, building a bearing model of:
wherein, K b For the stiffness matrix of the bearing, subscript b denotes the bearing, k ii ，
Respectively, the displacement stiffness coefficient of the support bearingAnd the angular stiffness coefficient, θ i For the translational direction rotational freedom of the system i, i belongs to (x, y, z), and the subscript x y z represents the translational freedom of the bearing in the first, second and third orientations of the system, respectively.
Wherein, the gear meshing model is constructed by the following steps:
as shown in FIG. 3, the meshing process of the gears can be simplified to a spring connection, i.e., k in the figure m (ii) a Without loss of generality, the meshing rigidity matrix of the helical gears is derived by taking the helical gears as an example in theoretical derivation. The stiffness matrix of the spur gear need only be such that the helix angle is zero. Thus, the relative displacement in the plane of engagement can be expressed as:
δ m ＝[(x p -x g )sinφ+(y p -y g )cosφ+(r p θ pz +r g θ gz )]cosβ
+[(r p θ py +r g θ gy )cosφ+(r p θ px +r g θ gx )sinφ+(z g -z p )]sinβ-e(t)
wherein, beta is a helical angle, phi is an end face pressure angle, and r represents a base circle radius; subscripts p, g denote the drive and driven gears, respectively; e (t) is the static transfer error, δ m Relative displacement in the direction of the meshing line; x, y and z respectively bearing the translational freedom degree of the system in the first direction, the second direction and the third direction, theta is the rotational freedom degree, and r is the rotational freedom degree p ,r g The base radius of the driving and driven gears are respectively.
Considering the gap nonlinearity, the meshing force between gear pairs can be expressed as:
wherein, F m Is the meshing force between gear pairs, k m For time-varying meshing stiffness, gamma m0 ，γ m1 Are all piecewise nonlinear clearance indicating functions, determined by the relative displacement in the direction of the meshing line and the backlash, gamma m0 For determining gear pairs under the effect of playElastic and damping forces of m1 The impact force between the gear pairs under the action of the clearance is determined; delta m Relative displacement in the direction of the meshing line; c. C m Is engaged damping; b is a clearance at the side of the half tooth,
is the meshing line relative velocity; wherein the content of the first and second substances,
in summary, the gear mesh model (equation of the meshing motion process) is:
wherein M is m ，C m ，K m Respectively a mass matrix, a meshing damping matrix and a meshing rigidity matrix;
q m acceleration, speed and displacement vector matrixes of the gear pair are respectively; e (t) is a static transfer error displacement function,
is a static transfer error velocity function; f. of m Is a coordinate transformation matrix;
f m ＝(c b s,cc b ,-s b ,r p ss b ,cr p s b ,c b r p ,-c b s,-cc b ,s b ,r g ss b ,cr g s b ,c b r g ) T ；
wherein, c b At a helix angleCosine value, s is the sine value of the pitch angle? c is the cosine of the end face pressure angle? s b The sine value of the end face pressure angle is shown, beta is a helical angle, phi is the end face pressure angle, and r represents the radius of a base circle; r is p ,r g Base radii of the driving and driven gears, c b ＝cosβ,s b ＝sinβ,c＝cosφ,s＝sinφ；
Wherein q is m ＝(x p ,y p ,z p ,θ px ,θ py ,θ pz ,x g ,y g ,z g ,θ gx ,θ gy ,θ gz ) T X, y and z respectively represent the translational freedom degrees of the gear in a first direction, a second direction and a third direction of the system, and theta is a rotational freedom degree; subscripts p, g represent driving and driven gears, respectively, and an upper subscript represents a transposition factor;
the construction of the superunit model specifically comprises the following steps:
as shown in fig. 4, when a three-dimensional model of a gear is built and a finite element model of the gear is built, generally, the number of grid nodes of the finite element model is large, calculation cannot be directly used, the degree of freedom of the finite element model needs to be reduced, a part of modal frequencies of the gear are reserved, and high-order frequencies of the gear are omitted. In order to couple with the beam unit node, a coupling node is established at the circle center of the inner ring of the gear, the coupling node and the inner ring of the gear are subjected to rigid constraint, the coupling node is set as a main degree of freedom, and an equivalent mass matrix and a rigidity matrix of the gear are obtained by utilizing a dynamic substructure method. And the equivalent mass matrix and the rigidity matrix of the gear are the superunit model.
Assembling the shaft unit model, the bearing model, the gear meshing model and the super unit model according to a gear transmission structure to obtain the gear transmission model; the method specifically comprises the following steps:
as shown in fig. 4, the 4 types of units are assembled into a mass matrix and a stiffness matrix according to the node numbers to obtain teeth
The overall mass and rigidity matrix of the wheel transmission system adopts proportional damping. The equation for the system is:
wherein, M, C and K respectively represent a mass matrix, a damping matrix and a rigidity matrix of the whole system, and F (t) represents external torque excitation and internal excitation.
Inputting the unit parameters of the shaft, the bearing parameters and the gear meshing parameters of the gear transmission system to be set into the traditional gear model, and solving the natural frequency of the gear transmission system, wherein the method comprises the following steps:
for undamped vibration with multiple degrees of freedom, the differential equation of motion can be written as:
the solution of the equation of the free vibration is simple harmonic vibration, and the solution of a certain simple harmonic vibration form in the vibration can be assumed as follows:
x＝φ i sinω i t
x&&＝-ω 2 φ i sinω i t
φ i is the magnitude of the amplitude, omega i Is the system vibration frequency. By substituting it into the free vibration equation, one can get:
([K]-ω i 2 [M])φ i ＝0
because of phi i For a non-zero vector, then the eigen equation can be obtained:
|([K]-ω i 2 [M])|＝0
wherein ω is i 2 I.e. the characteristic value, omega i Is the natural frequency of the system.
Solving the natural frequency of the gear transmission system with the natural frequency to be obtained is achieved through a matrix eigenvalue function, functions eig and eigs for solving the matrix eigenvalue are provided in matlab, the former is used for all eigenvalues of the matrix with few dimensions, and the latter can be used for solving the eigenvalue designated by the large matrix. Therefore, eigs is used in the present invention to solve, and the matrix eigenvalue function is specifically:
[V,D]＝eig(K,M)；
wherein, V is the obtained characteristic vector, D is the obtained characteristic value, K is the integral rigidity matrix, and M is the integral quality matrix.
And finally, calculating and setting the working rotating speed of the gear transmission system according to the natural frequency of the gear transmission system to be set, specifically setting the working rotating speed by avoiding the natural frequency.
Example three:
the invention will be described by taking a pair of spur gear transmission systems as an example; a three-dimensional model of the gear is shown in fig. 6. Establishing a gear concentrated mass model and a superunit model, calculating the natural frequencies of a gear shaft and a gear transmission system after assembling according to the steps, and comparing the natural frequencies with finite element results, wherein the comparison results are shown in tables 1-4:
TABLE 2 drive gear shaft natural frequency comparison
TABLE 3 driven gear shaft natural frequency comparison
TABLE 4 gear train natural frequency comparison
From the results in tables 1-4, it can be seen that simplifying the gears into a lumped mass model does not allow for the gear mode to be obtained, while reducing the natural frequency accuracy of the gear-shaft coupling. The gear superunit can be used for effectively improving the precision of a gear shaft model, obtaining more accurate natural frequency of a gear transmission system, and setting the working rotating speed according to the natural frequency, particularly setting the working rotating speed by avoiding the natural frequency.
In summary, according to the method and the system for setting the working speed of the gear transmission system, the superunit model of the gear wheel body is established by adopting a dynamic substructure method, the gear transmission model with higher precision is established by the superunit model of the gear wheel body, the natural frequency of the gear transmission system is solved by the gear transmission model, and the working speed of the gear is set according to the natural frequency. Because the gear transmission model is higher than the precision of the existing centralized quality model, the precision of the natural frequency solved according to the model with higher precision is more accurate, the resonance of the gear can be avoided better according to the working rotating speed of the gear set according to the natural frequency, the working safety of the gear is improved, and the technical problem that the actual working frequency is not avoided by the working rotating speed set according to the natural frequency with low precision, the gear generates resonance, and the service life of the gear is shortened is high.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (7)
1. A method for setting the operating speed of a gear transmission system, comprising the steps of:
obtaining shaft unit parameters, bearing parameters and meshing parameters of the gear in historical data, and respectively constructing a shaft unit model, a bearing model and a gear meshing model according to the shaft unit parameters, the bearing parameters and the meshing parameters; establishing a superunit model of the gear wheel body according to the shaft unit model, the bearing model and the gear meshing model; wherein, the super unit model of establishing the gear wheel body adopts the dynamic substructure method to accomplish, includes:
establishing a coupling node at the circle center of an inner ring of the gear, rigidly constraining the coupling node and the inner ring of the gear, setting the coupling node as a main degree of freedom, and obtaining an equivalent mass matrix and a stiffness matrix of the gear by a dynamic substructure method, wherein the equivalent mass matrix and the stiffness matrix of the gear are super-unit models;
assembling the shaft unit model, the bearing model, the gear meshing model and the super unit model according to a gear transmission structure to obtain a gear transmission model;
inputting the shaft unit parameters, the bearing parameters and the gear meshing parameters of the gear transmission system to be set into the traditional gear model, solving the natural frequency of the gear transmission system, and calculating and setting the working rotating speed of the gear transmission system according to the natural frequency.
2. The operating rotational speed setting method of a gear transmission system according to claim 1, wherein the shaft unit model is:
wherein, M S ，C s ，K S And F s Respectively a mass matrix, a damping matrix, a rigidity matrix and a load matrix of the shaft unit;
and q is S The acceleration matrix, the speed matrix and the displacement matrix of the shaft unit are respectively as follows:
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 ))；
wherein x, y and z are translation degrees of freedom of the axis unit in a first direction, a second direction and a third direction of the system respectively, the first direction, the second direction and the third direction are mutually vertical, i is the ith node of the axis unit, theta represents the rotation degree of freedom of the axis unit,
is an angular acceleration of the shaft unit,
is the angular velocity of the shaft unit.
3. The method for setting an operating rotational speed of a gear transmission system according to claim 2, wherein the bearing model is:
wherein, K b For the stiffness matrix of the bearing, subscript b denotes the bearing, k ii ，
Respectively, the displacement stiffness coefficient and the angular stiffness coefficient, theta, of the support bearing i For the rotational degrees of freedom in the translational direction of the system i, i e (x, y, z), the indices x, y and z represent the translational degrees of freedom of the bearing in the first, second and third orientations of the system, respectively.
4. The method for setting an operating speed of a gear transmission system according to claim 3, wherein the gear mesh model is:
wherein, F m Is the meshing force between gear pairs, k m For time-varying meshing stiffness, gamma m0 ，γ m1 Are all piecewise nonlinear clearance indicating functions, determined by the relative displacement in the direction of the meshing line and the backlash, gamma m0 For determining elastic and damping forces, gamma, between gear pairs under the influence of play m1 The impact force between the gear pairs under the action of the clearance is determined; delta m Relative displacement in the direction of the meshing line; c. C m Is engaged damping; b is a clearance at the side of the half tooth,
is the meshing line relative velocity; m m ，C m ，K m Respectively a mass matrix, a meshing damping matrix and a meshing rigidity matrix;
q m acceleration, speed and displacement vector matrixes of the gear pair are respectively; e (t) is a static transfer error displacement function,
is a static transfer error velocity function; f. of m Is a coordinate transformation matrix;
f m ＝(c b s,cc b ,-s b ,r p ss b ,cr p s b ,c b r p ,-c b s,-cc b ,s b ,r g ss b ,cr g s b ,c b r g ) T ；
wherein, c b Is the cosine of the helix angle, s is the sine of the helix angle, c is the cosine of the end face pressure angle, s b The sine value of the end face pressure angle, beta is a helical angle, phi is the end face pressure angle, and r represents the radius of a base circle; r is p ,r g The base radius of the driving gear and the driven gear are respectively;
c b ＝cosβ,s b ＝sinβ,c＝cosφ,s＝sinφ；
wherein q is m ＝(x p ,y p ,z p ,θ px ,θ py ,θ pz ,x g ,y g ,z g ,θ gx ,θ gy ,θ gz ) T X, y and z respectively represent the translation freedom degree of the system, and theta is the rotation freedom degree; subscripts p, g represent driving and driven gears, respectively, and an upper subscript represents a transposition factor;
wherein, the first and the second end of the pipe are connected with each other,
wherein the content of the first and second substances,
5. the method of setting an operating rotational speed of a gear transmission system according to claim 4, wherein the shaft unit model, the bearing model, the gear mesh model, and the superunit model are assembled in accordance with a gear transmission structure to obtain the gear transmission model; the method specifically comprises the following steps:
assembling the shaft unit model, the bearing model, the gear meshing model and the mass matrix and the rigidity matrix of the super unit model to obtain the mass matrix and the rigidity matrix of the whole gear transmission system; namely:
wherein M, C, K respectively represent the mass matrix, damping matrix, rigidity matrix of the whole system, f (q) represents the nonlinear displacement function caused by the clearance, and the nonlinear clearance indication function gamma is represented by the above function m0 It is decided that F (t) represents the external torque excitation and the internal excitation.
6. The method for setting the operating speed of the gear transmission system according to claim 5, wherein the solving of the natural frequency of the gear transmission system whose natural frequency is to be obtained is performed by a matrix eigenvalue function, wherein the matrix eigenvalue function is specifically:
[V,D]＝eig(K,M)；
wherein V is the solved eigenvector, D is the solved eigenvalue, K is the overall stiffness matrix, and M is the overall mass matrix.
7. A computer system comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the steps of the method of any one of claims 1 to 6 are performed when the computer program is executed by the processor.
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