CN110516384B - Multi-input multi-output gear transmission system dynamics modeling method - Google Patents
Multi-input multi-output gear transmission system dynamics modeling method Download PDFInfo
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Abstract
The invention discloses a dynamic modeling method for a multi-input multi-output gear transmission system, which comprises the following steps: s1 dividing the multiple-input multiple-output gear transmission system into a plurality of sub transmission systems; s2, respectively carrying out rigid body modeling on the components in each sub-transmission system and then assembling the components to form a subsystem model; s3, simplifying the coupler into a multi-degree-of-freedom spring with rigidity and damping, and connecting the subsystem models through the coupler to form an overall model of the multi-input multi-output gear transmission system; s4, carrying out multi-body dynamics simulation analysis on the overall model to obtain discrete data of wave dynamic force and wave dynamic moment at the position of the multi-degree-of-freedom spring; s5, carrying out numerical fitting on the discrete data to obtain a fluctuation function changing along with time; s6, establishing a concentrated mass dynamic model of each subsystem, and adding a fluctuation function to a degree of freedom equation of a corresponding node of the concentrated mass dynamic model to obtain the decoupled dynamic model of each subsystem. The invention has simple modeling and high solving speed.
Description
Technical Field
The invention belongs to the technical field of dynamics analysis, and particularly relates to a dynamics modeling method for a multi-input multi-output gear transmission system.
Background
The ship combined power transmission device is the core equipment of a marine ship, and is mainly formed by connecting a plurality of gear boxes in series or in parallel to form an integral gear transmission system, namely, the ship gear transmission system is a typical multi-input multi-output gear transmission system. The noise generated by the vibration of the gear transmission system can influence the radiation noise of the ship transmission device, so that how to reasonably establish a dynamic model of the ship gear transmission system is the basis for researching the vibration influence rule of the gear transmission system and optimizing the design of the ship gear transmission system.
The existing modeling method for the research of the gear system dynamics mainly comprises a concentrated mass method, a finite element method, multi-body dynamics, a statistical energy method and the like. The concentrated mass method is the most common method for modeling the gear transmission system, but the difficulty lies in extracting the concentrated parameters of each part because the gear transmission system is a mass distribution continuum; the finite element method divides a system into a gear unit, a shaft section unit, a bearing unit, a box body unit and the like, and establishes a system motion differential equation through a force balance condition and a displacement coordination equation of the units, and the method has high solving precision, however, when the scale of the gear transmission system is large, the calculation time is difficult to accept; the multi-body dynamics method has higher calculation efficiency than finite element calculation, but when gear meshing excitation is considered, the meshing stiffness is simplified into a spring-damping model, and the calculation accuracy is relatively poor; the statistical energy method is generally used for high-frequency analysis, can model simple-structure models such as plates and beams at present, and generally simplifies the models for complex gear transmission systems.
Aiming at a dynamic model of a complex gear transmission system, because the system configuration is complex, the number of parts forming the system is large, the degree of freedom and the excitation factors of the system are large, and a full-degree-of-freedom coupling model established based on a finite element method has the defects of low solving speed and complex modeling process under the condition of large excitation factors.
Disclosure of Invention
In view of this, the invention aims to provide a multiple-input multiple-output gear transmission system dynamics modeling method, which solves the defects caused by multiple degrees of freedom and multiple frequency excitation characteristics of the original overall system dynamics model, rigid modeling is performed on a complex system through the existing commercial software, boundary conditions of a coupling model are calculated, and then the boundary conditions are applied to a decoupled dynamics model based on a generalized finite element model, so that the calculation efficiency of system dynamics response is improved.
The technical scheme of the invention is as follows:
a multi-input multi-output gear transmission system dynamics modeling method comprises the following steps:
s1, dividing the multi-input multi-output gear transmission system into a plurality of sub transmission systems from the coupler according to the transmission relation;
s2, respectively carrying out rigid body modeling on the components in each sub-transmission system, and assembling the components to form a subsystem model according to the assembly relation;
s3, simplifying the coupler into a multi-degree-of-freedom spring with rigidity and damping, and connecting the subsystem models through the coupler to form an overall model of the multi-input multi-output gear transmission system;
s4, applying initial conditions to the overall model of the multi-input multi-output gear transmission system in a simulation environment, and carrying out multi-body dynamics simulation analysis on the overall model to obtain discrete data of wave power and wave power moment at the position of the multi-degree-of-freedom spring;
s5, carrying out numerical fitting on the discrete data of the wave power and the wave power moment to obtain a fluctuation function delta F (t) changing along with time;
s6, establishing a concentrated mass dynamic model of each subsystem, and adding a fluctuation function delta F (t) to a degree of freedom equation of a corresponding node of the concentrated mass dynamic model to obtain each decoupled subsystem dynamic model.
Preferably, the numerical fitting in S5 uses a least squares method, which includes the steps of:
s51 wave power and wave power moment in the form of discrete points at the multi-degree-of-freedom spring are expressed as
S52, using the polyfit function pair in MATLAB software
And performing linear fitting on the polynomial function to obtain a fitted fluctuation function delta F (t).
Preferably, the method for establishing the lumped mass dynamics model of each subsystem in S6 includes the following steps:
s61, establishing a dynamic model of each subsystem according to the generalized finite element theory, namely a full-freedom degree coupling dynamic model of the gear-shaft-bearing-box system, expressing an integral motion differential equation of the gear-shaft-bearing-box system by using a formula (1),
wherein,
m refers to a system overall quality matrix;
c refers to an integral damping matrix of the system;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 refers to the system external load vector;
s62, establishing a full-freedom coupling dynamic model of the gear-shaft-bearing-box system, which comprises the following steps:
s621, establishing a dynamic model of the input shaft and the output shaft, dividing the input shaft and the output shaft into a plurality of shaft sections, taking two ends of each shaft section as nodes, taking a unit formed by two adjacent nodes on the same shaft as a shaft section unit, and establishing the dynamic model of the shaft section unit shown in the formula (2) according to the Timoshenko beam principle
Wherein,
M s21 ＝M s12 ，M s22 ＝M s11
M s a quality matrix of the shaft segment unit;
C s a damping matrix which is a shaft section unit;
K s a stiffness matrix of the shaft section unit;
X s the displacement column vector of the shaft segment unit is obtained;
rho is the material density, kg/m 3 ；
A is the cross-sectional area of the cell, m 2 ；
l is the length of the unit, m;
j is polar moment of inertia, m 4 ；
E is the elastic modulus of the material, Pa;
g is a material shearing elastic model Pa;
a is the cross-sectional area of the cell, m 2 ；
L is the length of the unit, m;
I x is the second moment of area in the yz coordinate plane, m 4 ；
I y Is the second moment of area in the xz coordinate plane, m 4 ；
J is polar moment of inertia, m 4 ；
k is a correction factor, k is 10/9;
d is the outer diameter of the shaft section m;
d is the inner diameter of the shaft section, m;
s622, establishing a dynamic model of the driving gear and the driven gear, wherein the gear is equivalent to a cylinder, the outer diameter of the cylinder is the reference circle diameter of the gear, the inner diameter of the cylinder is the inner diameter of the gear hub, the meshing relation of the cylinder and the hub is represented by connecting two wheel bodies through a spring with time-varying meshing stiffness, and the rotational inertia I of the gear cylinder is calculated by using a formula (3) s ，
Wherein,
D 1 is the outer diameter, m;
d 1 is the inner diameter, m;
L 1 is the length of the equivalent cylinder, m;
rho is the material density, kg/m 3 ；
Superposing the rotational inertia of the gear cylinder to a corresponding node of a shaft connected with the gear, and calculating the time-varying meshing rigidity of the gear according to ISO-6366 and the length of an instantaneous contact line;
s623, establishing a bearing unit dynamic model, neglecting the mass and inertia of a bearing, simplifying the bearing into a spring with rigidity and damping, and measuring the rigidity value and the damping value according to an experiment;
s624, establishing a dynamic model of the box body unit, calculating the gear box body by adopting a substructure method, arranging a main node in the center of a bearing hole of the box body, and extracting an equivalent stiffness matrix K of the box body condensed at the main node g Sum quality matrix M g Building a tank using equation (4)A model of the dynamics of the body is described,
wherein,
C g ＝α 0 M g +α 1 K g
M g a quality matrix of the box unit;
C g a damping matrix which is a box unit;
K g a stiffness matrix of the box unit;
X g the displacement column vector of the box body unit is obtained;
α 0 is the mass proportionality coefficient in Rayleigh damping;
α 1 is the stiffness proportionality coefficient in Rayleigh damping;
s625, assembling each mass matrix, each rigidity matrix and each damping matrix according to the interconnection relationship of parts in the gear system, simplifying a driving gear and a driven gear into a node on a shaft connected with the driving gear and the driven gear, combining the node with the node on the original shaft, superposing inertia, connecting one end of a bearing unit with the node of the shaft mounting position of the bearing and connecting one end of the bearing unit with the node of the bearing hole of the box unit, and superposing sub-matrices of different unit types to the corresponding position of the whole matrix according to the corresponding relationship among the nodes to form a dynamic model of the system;
s63, adding the obtained fluctuation function at the joint of the coupling spring to the degree of freedom of the corresponding joint of the kinetic equation to obtain the integral motion differential equation of the subsystem shown in the following formula (5)
Wherein,
m refers to the system overall quality matrix;
c refers to the system integral damping matrix;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 a system external load vector;
Δ f (t) refers to the ripple function column vector at the joint of the coupling spring.
Preferably, the damping matrix C in step S621 s Rayleigh damping form determination using equation (6),
C m ＝α 0 M m +α 1 k m ,m＝1,2… (6)
in the formula:
C m the damping coefficient of the mth order vibration mode;
α 0 is the mass proportionality coefficient in Rayleigh damping;
α 1 is the stiffness proportionality coefficient in Rayleigh damping;
M m is the mode mass of the mth order mode;
K m is the stiffness of the mth order mode;
and m is the order of mode shape.
Compared with the prior art, the multi-input multi-output gear transmission system dynamics modeling method provided by the invention has the beneficial effects that:
1. the invention establishes a decoupled multi-input multi-output gear transmission system dynamic model, improves the calculation efficiency of system dynamic response and can ensure certain calculation precision;
2. the model obtained based on the multi-body dynamics method has high visualization degree and is easy to modify;
3. the invention can reduce the operation difficulty of researchers and save manpower;
4. the invention has good practicability and is worth popularizing.
Drawings
FIG. 1 is a flow chart of a method for modeling dynamics of a multiple-input multiple-output gear transmission system;
FIG. 2 is a schematic diagram of a dual input, single output transmission configuration;
FIG. 3 is a schematic diagram of a multi-rigid-body model of a G1(G2) subsystem;
FIG. 4 is a schematic diagram of a multi-rigid-body model of the G3 subsystem;
FIG. 5 is a schematic diagram of a G1 system dynamics model.
Detailed Description
The invention provides a dynamic modeling method for a multi-input multi-output gear transmission system, which is described in the following by combining with a flow schematic diagram of figure 1.
As shown in fig. 1, the technical solution of the present invention is:
a multi-input multi-output gear transmission system dynamics modeling method comprises the following steps:
s1, dividing the multi-input multi-output gear transmission system into a plurality of sub transmission systems from the coupler according to the transmission relation;
s2, respectively performing rigid body modeling on the components in each sub-transmission system, and assembling the components according to the assembly relationship to form a subsystem model;
s3, simplifying the coupler into a multi-degree-of-freedom spring with rigidity and damping, and connecting the subsystem models through the coupler to form an overall model of the multi-input multi-output gear transmission system;
s4, applying initial conditions to the overall model of the multi-input multi-output gear transmission system in a simulation environment, and carrying out multi-body dynamics simulation analysis on the overall model to obtain discrete data of wave power and wave power moment at the position of the multi-degree-of-freedom spring;
s5, carrying out numerical fitting on the discrete data of the wave power and the wave power moment to obtain a fluctuation function delta F (t) changing along with time;
s6, establishing a concentrated mass dynamics model of each subsystem, and adding a fluctuation function delta F (t) to a degree of freedom equation of a corresponding node of the concentrated mass dynamics model to obtain each decoupled subsystem dynamics model.
Further, the numerical fitting in S5 adopts a least square method, which includes the following steps:
s51 wave power and wave power moment in the form of discrete points at the multi-degree-of-freedom spring are expressed as
S52, using the polyfit function pair in MATLAB software
And performing linear fitting on the polynomial function to obtain a fitted fluctuation function delta F (t).
Further, the method for establishing the lumped mass dynamics model of each subsystem in S6 includes the following steps:
s61, establishing a dynamic model of each subsystem, namely a full-freedom degree coupling dynamic model of the gear-shaft-bearing-box system according to the generalized finite element theory, representing the integral motion differential equation of the gear-shaft-bearing-box system by using a formula (1),
wherein,
m refers to the system overall quality matrix;
c refers to the system integral damping matrix;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 refers to the system external load vector;
s62, establishing a full-freedom coupling dynamic model of the gear-shaft-bearing-box body system, which specifically comprises the following steps:
s621, establishing a dynamic model of the input shaft and the output shaft, dividing the input shaft and the output shaft into a plurality of shaft sections, wherein two ends of each shaft section are nodes, a unit formed by two adjacent nodes on the same shaft is a shaft section unit, and establishing the dynamic model of the shaft section unit as shown in the formula (2) according to the Timoshenko beam principle
Wherein,
M s21 ＝M s12 ，M s22 ＝M s11
M s a quality matrix of the shaft segment unit;
C s a damping matrix which is a shaft section unit;
K s a stiffness matrix of the shaft section unit;
X S the displacement column vector of the shaft segment unit is obtained;
rho is the material density, kg/m 3 ；
A is the cross-sectional area of the cell, m 2 ；
l is the length of the unit, m;
j is polar moment of inertia, m 4 ；
E is the elastic modulus of the material, Pa;
g is a material shearing elastic model Pa;
a is the cross-sectional area of the cell, m 2 ；
L is the length of the unit, m;
I x is the cross-sectional moment of inertia, m, in the yz coordinate plane 4 ；
I y Is the second moment of area in the xz coordinate plane, m 4 ；
J is polar moment of inertia, m 4 ；
k is a correction factor, k is 10/9;
d is the outer diameter of the shaft section m;
d is the inner diameter of the shaft section, m;
s622, establishing a dynamic model of the driving gear and the driven gear, wherein the gear is equivalent to a cylinder, the outer diameter of the cylinder is the reference circle diameter of the gear, the inner diameter of the cylinder is the inner diameter of the gear hub, the meshing relation of the cylinder and the gear hub is represented by connecting two wheel bodies through a spring with time-varying meshing stiffness, and the rotational inertia I of the gear cylinder is calculated by using the formula (3) s ，
Wherein,
d is the outer diameter (m);
d is the inner diameter (m);
l is the length (m) of the equivalent cylinder;
rho is the material density (kg/m) 3 )；
Superposing the rotational inertia of the gear cylinder to a corresponding node of a shaft connected with the gear, and calculating the time-varying meshing rigidity of the gear according to ISO-6366 and the length of an instantaneous contact line;
s623, establishing a bearing unit dynamic model, neglecting the mass and inertia of a bearing, simplifying the bearing into a spring with rigidity and damping, and measuring the rigidity value and the damping value according to an experiment;
s624, establishing a dynamic model of the box body unit, calculating the gear box body by adopting a substructure method, arranging a main node in the center of a bearing hole of the box body, and extracting an equivalent stiffness matrix K of the box body condensed at the main node g Sum quality matrix M g The dynamic model of the box body is established by using the formula (4)
Wherein,
C g ＝α 0 M g +α 1 K g
M g a quality matrix of the box unit;
C g a damping matrix which is a box unit;
K g a stiffness matrix of the box unit;
X g the displacement column vector of the box body unit is obtained;
α 0 is the mass proportionality coefficient in Rayleigh damping;
α 1 is the stiffness proportionality coefficient in Rayleigh damping;
s625, assembling each mass matrix, each rigidity matrix and each damping matrix according to the interconnection relationship of parts in the gear system, simplifying a driving gear and a driven gear into a node on a shaft connected with the driving gear and the driven gear, combining the node with the node on the original shaft, superposing inertia, connecting one end of a bearing unit with the node of the shaft mounting position of the bearing and connecting one end of the bearing unit with the node of the bearing hole of the box unit, and superposing sub-matrices of different unit types to the corresponding position of the whole matrix according to the corresponding relationship among the nodes to form a dynamic model of the system;
s63, adding the obtained fluctuation function at the joint of the coupling spring to the degree of freedom of the corresponding joint of the kinetic equation to obtain the integral motion differential equation of the subsystem shown in the following formula (5)
Wherein,
m refers to the system overall quality matrix;
c refers to an integral damping matrix of the system;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 a system external load vector;
Δ f (t) refers to the ripple function column vector at the joint of the coupling spring.
Further, the damping matrix C in step S621 s Rayleigh damping form determination using equation (6),
C m ＝α 0 M m +α 1 k m (m＝1,2…) (6)
in the formula:
C m refers to the damping coefficient of the mth order mode;
α 0 is the mass proportionality coefficient in Rayleigh damping;
α 1 is the stiffness proportionality coefficient in Rayleigh damping;
M m is the mode mass of the mth order mode;
K m is the stiffness of the mth order mode;
and m is the order of mode shape.
Example 1
The embodiment takes a dual-input single-output gear transmission system as an example, and introduces a specific implementation manner of the invention, which mainly comprises the following steps:
s1, dividing the multi-input multi-output gear transmission system into a plurality of sub transmission systems from the coupler according to the transmission relation;
s2, respectively carrying out rigid body modeling on the components in each sub-transmission system, and assembling the components to form a subsystem model according to the assembly relation;
s3, simplifying the coupler into a multi-degree-of-freedom spring with rigidity and damping, and connecting the subsystem models through the coupler to form an overall model of the multi-input multi-output gear transmission system;
s4, applying initial conditions to the overall model of the multi-input multi-output gear transmission system in a simulation environment, and carrying out multi-body dynamics simulation analysis on the overall model to obtain discrete data of wave power and wave power moment at the position of the multi-degree-of-freedom spring;
s5, carrying out numerical fitting on the wave power and the discrete data of the wave power moment to obtain a wave function delta F (t) changing along with time;
s6, establishing a concentrated mass dynamic model of each subsystem, and adding a fluctuation function delta F (t) to a degree of freedom equation of a corresponding node of the concentrated mass dynamic model to obtain each decoupled subsystem dynamic model.
The step S1 specifically includes the following steps:
in connection with an embodiment, as shown in fig. 2, the dual-input single-output gear transmission system is composed of three gear box transmission system groups connected by couplings, thus dividing the overall gear transmission system into three sub-transmission systems: a G1 subsystem, a G2 subsystem, and a G3 subsystem.
The step S2 specifically includes the following steps:
the parts of each sub-transmission system mainly comprise a gear, an input shaft, an output shaft, a bearing and a box body, and the gear, the input shaft and the box body are respectively modeled according to the known parameters and dimensions of the parts and then assembled according to the assembly relationship to form the overall sub-system.
Fig. 3 shows a schematic diagram of a multi-rigid-body model of a G1 subsystem, which comprises a G1 gear box 1, a G1 driven gear 2, a G1 output shaft 3, a G1 input shaft 4 and a G1 driving gear 5.
The G1 driving gear 5 is fixedly connected with a G1 input shaft 4, a G1 driven gear 2 is fixedly connected with a G1 output shaft 3, the G1 driving gear 5 is connected with the G1 driven gear 2 through a spring with periodic time-varying stiffness, the G1 input shaft 4, the G1 output shaft 3 and the G1 gearbox 1 are connected through a multi-degree-of-freedom spring simplified by a bearing, one end of the spring is connected to the center point of a bearing hole of the gearbox, and the other end of the spring is connected to the center of the mounting position of the bearing on the shaft, and the steps are repeated to obtain a multi-rigid-body dynamic model of a G1 subsystem.
The multi-rigid-body dynamic model of the G2 subsystem is structurally the same as the multi-rigid-body dynamic model of the G1 subsystem.
Fig. 4 shows a schematic diagram of a multi-rigid-body model of a G3 subsystem, which comprises a G3 gear box 6, a G3 driving gear 7, a G3 input shaft 8, a G3 intermediate shaft 9, a G3 intermediate reduction gear 10, a G3 output shaft 11, and a G3 driven gear 12.
The G3 driving gear 7 is fixedly connected with a G3 input shaft 8, a G3 intermediate reduction gear 10 is fixedly connected with a G3 intermediate shaft 9, a G3 driven gear 12 is fixedly connected with a G3 output shaft 11, a G3 driving gear 7 is connected with the G3 intermediate reduction gear 10, the G3 intermediate reduction gear 10 and a G3 driven gear 12 through springs with periodic time-varying stiffness, the G3 input shaft 8, the G3 intermediate shaft 9, the G3 output shaft 11 and a G3 gear box body 6 are connected through a multi-degree-of-freedom spring with simplified bearings, one end of each spring is connected to the center point of a bearing hole of the box body, and the other end of each spring is connected to the center point of the mounting position of the bearing on the shaft, so that a multi-dynamic model of a G3 subsystem is obtained.
The step S3 specifically includes the following steps:
equally dividing the mass and the rotational inertia of the coupler and adding the mass and the rotational inertia to a shaft section connected with the coupler; then the coupler is simplified into a multi-degree-of-freedom spring with rigidity and damping, and the mathematical expression form of the multi-degree-of-freedom spring is as follows:
in the formula:
F x 、F y 、F z respectively representing the component force of the transmission force of the coupler in the directions of x, y and z;
T α 、T β 、T γ respectively representing the partial moments of the coupling transmission moment in the directions of x, y and z;
x, Y, Z, alpha, beta, gamma denote the displacement and angular displacement of the coupling;
K i (i ═ x, xy …) represents the coupling stiffness values;
C i (i ═ x, xy …) represents the damping values of the couplings;
F x1 、F y1 、F z1 respectively representing the component forces of the nominal force of the coupling in the directions of x, y and z;
T α 、T β 、T γ respectively representing the partial moments of the nominal moment of the coupling in the directions around x, y and z.
A coupling spring is coupled between the output shaft node of the G1 system and the input shaft node of the G3 system, and a coupling spring is coupled between the G2 output shaft node and the G3 input shaft node.
Further, the node selection principle is as follows: and selecting central points of coupling shaft sections of the coupling and the transmission shaft of the gearbox in the actual configuration of the system, wherein at the moment, all subsystems are coupled to form an overall double-input single-output rigid body model.
The step S4 specifically includes the following steps:
in combination with the actual working conditions of double input and single output, corresponding rotating speed and power are respectively applied to input shafts of a G1 system and a G2 system, load torque is applied to an output shaft of the G3 system, and the rotating speed of an output end is calculated according to the following gear transmission expression:
in the formula:
n out is the output rotation speed;
n in is the input rotation speed;
z 1 …z i the number of teeth of the driving gear;
z 2 …z n the number of teeth of the driven gear.
Specific values of a rigidity matrix and a damping matrix of the bearing and the coupler are measured through experiments;
the meshing rigidity of the gears is obtained according to ISO-6366 and the length of the instantaneous contact line, and the obtained time-varying meshing rigidity is added to a gear meshing unit in the overall model of the multi-input multi-output gear transmission system.
And carrying out multi-body dynamics analysis on the established double-input single-output gear transmission system to obtain the wave power and the wave power moment at the spring node of the coupler.
The numerical fitting in the step S5 specifically includes the following steps by using a least square method:
s51 wave power and wave power moment in the form of discrete points at the multi-degree-of-freedom spring are expressed as
S52, using the polyfit function pair in MATLAB software
And performing linear fitting on the polynomial function to obtain a fitted fluctuation function delta F (t).
The step S6 specifically includes the following steps:
and S61, establishing a dynamic model of each subsystem, namely a full-freedom coupling dynamic model of the gear-shaft-bearing-box system according to the generalized finite element theory. As shown in fig. 5, taking the G1 subsystem as an example, the subsystems are separated into different types of units and nodes, and the unit types specifically include a bearing spring unit 13, a driven gear unit 14, a coupling unit 15, a gear box unit 16, a driving gear unit 17, a shaft segment unit 18, a meshing unit 19 and a coupling spring unit 20.
The driving gear unit 17 is fixedly connected with the shaft section unit 18 through a connecting node, the driven gear unit 14 is fixedly connected with the shaft section unit 18 through a connecting node, the driving gear unit 17 is connected with the driven gear unit 14 through an engaging unit 19, the coupling unit 15 is connected with the shaft section unit 18 through a coupling spring unit 20, the shaft section unit 18 is connected with the gear box unit 16 through a bearing spring unit 13, one end of the spring is connected to a bearing hole node of the box, and the other end of the spring is connected to a mounting position node of a bearing on a shaft, and the steps are repeated to obtain the multi-rigid-body dynamic model of the G1 subsystem.
The multi-rigid-body dynamic model of the G2 subsystem has the same structure as the multi-rigid-body dynamic model of the G1 subsystem.
Then, each unit is subjected to stress analysis, a corresponding motion differential equation is established, and the mass, rigidity and damping matrix of each unit is assembled according to a finite element method to obtain the system integral differential equation which takes the node displacement of each unit as a generalized coordinate as shown in the specification
In the above formula, the first and second carbon atoms are,
m refers to a system overall quality matrix;
c refers to an integral damping matrix of the system;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 refers to the system external load vector.
S62, establishing a full-freedom coupling dynamic model of the gear-shaft-bearing-box body system specifically comprises the following steps:
s621, establishing a dynamic model of the input shaft and the output shaft, dividing the input shaft and the output shaft into a plurality of shaft sections, wherein two ends of each shaft section are nodes, a unit formed by two adjacent nodes on the same shaft is a shaft section unit, and establishing the dynamic model of the shaft section unit as shown in the specification according to the Timoshenko beam principle
Wherein,
M s21 ＝M s12 ，M s22 ＝M s11
M s a quality matrix of the shaft segment unit;
C s a damping matrix which is a shaft section unit;
K s a stiffness matrix of the shaft section unit;
X s the displacement column vector of the shaft segment unit is obtained;
rho is material density, kg/m 3 ；
A is the cross-sectional area m of the cell 2 ；
l is the length of the unit, m;
j is polar moment of inertia, m 4 ；
E is the elastic modulus of the material, Pa;
g is a material shearing elastic model Pa;
a is the cross-sectional area of the cell, m 2 ；
L is the length of the unit, m;
I x is the cross-sectional moment of inertia, m, in the yz coordinate plane 4 ；
I y Is the second moment of area in the xz coordinate plane, m 4 ；
J is polar moment of inertia, m 4 ；
k is a correction factor, k is 10/9;
d is the outer diameter of the shaft section, m;
d is the shaft section inner diameter, m.
S622, establishing a dynamic model of the driving gear and the driven gear, wherein the gear is equivalent to a cylinder, the outer diameter of the cylinder is the reference circle diameter of the gear, the inner diameter of the cylinder is the inner diameter of the gear hub, the meshing relation of the cylinder and the gear hub is represented by connecting two gear bodies through a spring with time-varying meshing stiffness, and the rotational inertia I of the gear cylinder is calculated by the following formula s ，
Wherein,
D 1 is the outer diameter, m;
d 1 is the inner diameter, m;
L 1 is the length of the equivalent cylinder, m;
rho is the material density, kg/m 3 ；
And (3) superposing the rotational inertia of the gear cylinder to a corresponding node of the shaft connected with the gear, wherein the time-varying meshing rigidity of the gear is calculated according to ISO-6366 and the length of the instantaneous contact line.
S623, establishing a bearing unit dynamic model, neglecting the mass and inertia of a bearing, simplifying the bearing into a multi-degree-of-freedom spring with rigidity and damping, wherein the rigidity value and the damping value are measured according to experiments, and the mathematical expression form is as follows:
in the formula:
F x 、F y 、F z respectively representing the component force of the bearing force in the directions of x, y and z;
T α 、T β 、T γ respectively representing the partial moments of the bearing moment in the directions of x, y and z;
x, Y, Z, alpha, beta, gamma represent displacements and angular displacements of the bearing in different directions;
K i (i ═ x, xy …) represents the stiffness values of the bearing;
C i and (i ═ x, xyi) represents various damping values of the bearing.
S624, establishing a dynamic model of the box body unit, calculating the gear box body by adopting a substructure method, arranging a main node in the center of a bearing hole of the box body, and extracting an equivalent stiffness matrix K of the box body condensed at the main node g Sum quality matrix M g The dynamic model of the box body is established by the following formula,
wherein,
C g ＝α 0 M g +α 1 K g
M g a quality matrix of the box unit;
C g a damping matrix which is a box body unit;
K g a stiffness matrix that is a box unit;
X g a displacement column vector of the box body unit;
α 0 is the mass proportionality coefficient in Rayleigh damping;
α 1 is the stiffness proportionality coefficient in Rayleigh damping;
s625, assembling each mass matrix, stiffness matrix and damping matrix according to the interconnection relationship of parts in the gear system, simplifying a driving gear and a driven gear into a node on a shaft connected with the driving gear and the driven gear, combining the node with the node on the original shaft, superposing inertia, connecting one end of a bearing unit with the node of the shaft installation position of a bearing and connecting one end of the bearing unit with the node of the bearing hole of a box unit, and superposing sub-matrices of different unit types to the corresponding position of the whole matrix according to the corresponding relationship between the nodes to form a dynamic model of the system;
s63, adding the obtained fluctuation function at the joint of the coupling spring to the degree of freedom of the corresponding joint of the kinetic equation to obtain the integral motion differential equation of the subsystem shown as follows
Wherein,
m refers to the system overall quality matrix;
c refers to the system integral damping matrix;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 a system external load vector;
Δ f (t) refers to the ripple function column vector at the joint of the coupling spring.
The method establishes the decoupled multi-input multi-output gear transmission system dynamic model, improves the calculation efficiency of system dynamic response, can ensure certain calculation precision, simultaneously has high degree of visualization of the model obtained based on a multi-body dynamic method, is easy to modify, can reduce the operation difficulty of researchers, saves manpower, has good practicability and is worthy of popularization.
The above disclosure is only for the preferred embodiments of the present invention, but the embodiments of the present invention are not limited thereto, and any variations that can be made by those skilled in the art are intended to fall within the scope of the present invention.
Claims (4)
1. A multi-input multi-output gear transmission system dynamics modeling method is characterized by comprising the following steps:
s1, dividing the multi-input multi-output gear transmission system into a plurality of sub transmission systems from the coupler according to the transmission relation;
s2, respectively carrying out rigid body modeling on the components in each sub-transmission system, and assembling the components to form a subsystem model according to the assembly relation;
s3, simplifying the coupler into a multi-degree-of-freedom spring with rigidity and damping, and connecting the subsystem models through the coupler to form an overall model of the multi-input multi-output gear transmission system;
s4, applying initial conditions to the overall model of the multi-input multi-output gear transmission system in a simulation environment, and carrying out multi-body dynamics simulation analysis on the overall model to obtain discrete data of wave power and wave power moment at the position of the multi-degree-of-freedom spring;
s5, carrying out numerical fitting on the discrete data of the wave power and the wave power moment to obtain a fluctuation function delta F (t) changing along with time;
s6, establishing a concentrated mass dynamics model of each subsystem, and adding a fluctuation function delta F (t) to a degree of freedom equation of a corresponding node of the concentrated mass dynamics model to obtain each decoupled subsystem dynamics model.
2. The modeling method of dynamics of a multiple-input multiple-output gear transmission system according to claim 1, wherein the numerical fitting in S5 employs a least squares method, which includes the steps of:
s51 wave power and wave power moment in the form of discrete points at the multi-degree-of-freedom spring are expressed as
S52, using the polyfit function pair in MATLAB software
And performing linear fitting on the polynomial function to obtain a fitted fluctuation function delta F (t).
3. The modeling method of dynamics of a multiple-input multiple-output gear transmission system according to claim 1, wherein the method of establishing a lumped mass dynamics model of each subsystem in S6 comprises the steps of:
s61, establishing a dynamic model of each subsystem according to the generalized finite element theory, namely a full-freedom degree coupling dynamic model of the gear-shaft-bearing-box system, expressing an integral motion differential equation of the gear-shaft-bearing-box system by using a formula (1),
wherein,
m refers to the system overall quality matrix;
c refers to an integral damping matrix of the system;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 refers to the system external load vector;
s62, establishing a full-freedom coupling dynamic model of the gear-shaft-bearing-box body system, which specifically comprises the following steps:
s621, establishing a dynamic model of the input shaft and the output shaft, dividing the input shaft and the output shaft into a plurality of shaft sections, wherein two ends of each shaft section are nodes, a unit formed by two adjacent nodes on the same shaft is a shaft section unit, and establishing the dynamic model of the shaft section unit as shown in the formula (2) according to the Timoshenko beam principle
Wherein,
M s21 ＝M s12 ，M s22 ＝M s11
M s a quality matrix of the shaft segment unit;
C s a damping matrix which is a shaft section unit;
K s a stiffness matrix of the shaft section unit;
X S the displacement column vector of the shaft segment unit is obtained;
rho is the material density, kg/m 3 ；
A is the cross-sectional area of the cell, m 2 ；
l is the length of the unit, m;
j is polar moment of inertia, m 4 ；
E is the elastic modulus of the material, Pa;
g is a material shearing elastic model Pa;
a is the cross-sectional area of the cell, m 2 ；
L is the length of the unit, m;
I x is the second moment of area in the yz coordinate plane, m 4 ；
I y Is the second moment of area in the xz coordinate plane, m 4 ；
J is polar moment of inertia, m 4 ；
k is a correction factor, k is 10/9;
d is the outer diameter of the shaft section, m;
d is the inner diameter of the shaft section, m;
s622, establishing a dynamic model of the driving gear and the driven gear, wherein the gear is equivalent to a cylinder, the outer diameter of the cylinder is the reference circle diameter of the gear, the inner diameter of the cylinder is the inner diameter of the gear hub, the meshing relation of the cylinder and the gear hub is represented by connecting two wheel bodies through a spring with time-varying meshing stiffness, and the rotational inertia I of the gear cylinder is calculated by using the formula (3) s ，
Wherein,
D 1 is the outer diameter, m;
d 1 is the inner diameter, m;
L 1 is the length of the equivalent cylinder, m;
rho is the material density, kg/m 3 ；
Superposing the rotational inertia of the gear cylinder to a corresponding node of a shaft connected with the gear and the shaft, wherein the time-varying meshing rigidity of the gear is calculated according to ISO-6366 and the length of an instantaneous contact line;
s623, establishing a bearing unit dynamic model, neglecting the mass and inertia of a bearing, simplifying the bearing into a spring with rigidity and damping, and measuring the rigidity value and the damping value according to an experiment;
s624, establishing a dynamic model of the box body unit, calculating the gear box body by adopting a substructure method, arranging a main node in the center of a bearing hole of the box body, and extracting an equivalent stiffness matrix K of the box body condensed at the main node g And a quality matrix M g A dynamic model of the box body is established by using the formula (4),
wherein,
C g ＝α 0 M g +α 1 K g
M g a quality matrix of the box unit;
C g a damping matrix which is a box body unit;
K g a stiffness matrix that is a box unit;
X g the displacement column vector of the box body unit is obtained;
α 0 is the mass proportionality coefficient in Rayleigh damping;
α 1 is the stiffness proportionality coefficient in Rayleigh damping;
s625, assembling each mass matrix, stiffness matrix and damping matrix according to the interconnection relationship of parts in the gear system, simplifying a driving gear and a driven gear into a node on a shaft connected with the driving gear and the driven gear, combining the node with the node on the original shaft, superposing inertia, connecting one end of a bearing unit with the node of the shaft installation position of a bearing and connecting one end of the bearing unit with the node of the bearing hole of a box unit, and superposing sub-matrices of different unit types to the corresponding position of the whole matrix according to the corresponding relationship between the nodes to form a dynamic model of the system;
s63, adding the obtained fluctuation function at the joint of the coupling spring to the degree of freedom of the corresponding joint of the kinetic equation to obtain the integral motion differential equation of the subsystem shown in the following formula (5)
Wherein,
m refers to the system overall quality matrix;
c refers to the system integral damping matrix;
k (t) refers to the overall stiffness matrix of the system;
x (t) refers to all node displacement column vectors;
P 0 refers to the system external load vector;
Δ f (t) refers to the ripple function column vector at the joint of the coupling spring.
4. The method of modeling multiple input multiple output gear train dynamics according to claim 3 wherein said damping matrix C of step S621 s Rayleigh damping form determination using equation (6),
C m ＝α 0 M m +α 1 K m ,m＝1,2… (6)
in the formula:
C m refers to the damping coefficient of the mth order mode;
α 0 is the mass proportionality coefficient in Rayleigh damping;
α 1 is the stiffness proportionality coefficient in Rayleigh damping;
M m is the mode mass of the mth order mode;
K m is the stiffness of the mth order mode;
and m is the order of mode shape.
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