CN110516384A - A kind of multiple-input and multiple-output gear train assembly dynamic modeling method - Google Patents
A kind of multiple-input and multiple-output gear train assembly dynamic modeling method Download PDFInfo
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Abstract
The invention discloses a kind of multiple-input and multiple-output gear train assembly dynamic modeling methods, comprising the following steps: multiple-input and multiple-output gear train assembly is divided into multiple sub-drivers and united by S1；S2 is assembled to form subsystem model after carrying out rigid body modeling respectively to the composition components in each sub-driver system；Shaft coupling is reduced to the multiple degrees of freedom spring with rigidity and damping by S3, is connected between each subsystem model by shaft coupling, is constituted the overall model of multiple-input and multiple-output gear train assembly；S4 carries out Simulation Analysis on Multi-body Dynamics to overall model, obtains the discrete data of wave force and fluctuation moment at multiple degrees of freedom spring；S5 carries out numerical fitting, the wave function changed over time to discrete data；S6 establishes the lumped mass kinetic model of each subsystem, wave function is added in the freedom degree equation of lumped mass kinetic model corresponding node, each subsystem kinetic model after being decoupled.Present invention modeling is simple, and solving speed is fast.
Description
Technical field
The invention belongs to Dynamics analysis technology fields, and in particular to a kind of multiple-input and multiple-output gear train assembly power
Learn modeling method.
Background technique
Combined marine actuating unit is the core equipment of marine ship, it mainly by multiple gear-boxes by series connection or
Mode in parallel connects into overall gear transmission system, that is to say, that ship gear train assembly is that a kind of typical multi input is more
Output gear transmission system.The noise that gear train assembly vibration generates will affect the radiated noise of marine transmission, therefore
And the kinetic model of ship gear train assembly how is rationally established, it is the base for studying gear train assembly vibration affecting laws
Plinth, and optimize the basis of the design of ship gear train assembly.
The existing modeling method for Gear system dynamic research mainly has concentrated quality method, FInite Element, more bodies
Dynamics, statistical Energy Analysis Approach etc..Concentrated quality method is the most common method of gear train assembly modeling, but due to pinion unit
System is Mass Distribution non-individual body, so difficult point is the extraction of the lumped parameter of each components；System is divided by FInite Element
Gear unit, shaft part unit, bearing unit, box unit etc. pass through the dynamic balance condition and displacement coordination establishing equation of unit
The system motion differential equation, this method solving precision with higher, however when gear train assembly is in large scale, it calculates
Time is difficult to receive；Many-body dynamics method has computational efficiency more higher than FEM calculation, however it is nibbled in consideration gear
When closing excitation, mesh stiffness is reduced to spring-damper model, computational accuracy is relatively poor；Statistical Energy Analysis Approach is generally used for height
Frequency analysis at present can would generally carry out complicated gear transmission system the simple model modeling of the structures such as plate, beam to model
It is a large amount of to simplify.
For the kinetic model of complicated gear transmission system, due to system configuration complexity, form the components of system compared with
More, the number of degrees of freedom, and motivator of system are more, based on FInite Element establish full Degree-of-freedom Coupling model excitation because
Slow, the disadvantage of modeling process complexity that there are solving speeds in the case that element is more.
Summary of the invention
In view of this, the object of the present invention is to provide a kind of multiple-input and multiple-output gear train assembly Dynamic Modeling sides
Method, the drawbacks of being brought this method solve original total system kinetic model due to multiple degrees of freedom and multi-frequency excitation feature,
Rigid body modeling is carried out to complication system by existing business software, the boundary condition of coupling model is calculated, is re-applied to decouple
In the kinetic model based on generalized finite meta-model afterwards, the computational efficiency of system dynamics response, while base are improved
It is high in the model visualization degree that many-body dynamics method obtains, it is easy to modify, researcher's operation difficulty can be reduced, saves people
Power.
The technical scheme is that
A kind of multiple-input and multiple-output gear train assembly dynamic modeling method, comprising the following steps:
S1, multiple-input and multiple-output gear train assembly is divided by multiple sub-drivers from shaft coupling according to drive connection
System；
S2, rigid body modeling is carried out to the composition components in each sub-driver system respectively, is assembled according to assembly relation
Form subsystem model；
S3, shaft coupling is reduced to the multiple degrees of freedom spring with rigidity and damping, passes through connection between each subsystem model
Axis device is connected, and constitutes the overall model of multiple-input and multiple-output gear train assembly；
S4, primary condition is applied to the overall model of multiple-input and multiple-output gear train assembly in a simulated environment, to it
Simulation Analysis on Multi-body Dynamics is carried out, the discrete data of wave force and fluctuation moment at multiple degrees of freedom spring is obtained；
S5, numerical fitting, the wave function changed over time are carried out to the discrete data of wave force and fluctuation moment
ΔF(t)；
It is dynamic to be added to lumped mass by S6, the lumped mass kinetic model for establishing each subsystem by wave function Δ F (t)
Each subsystem kinetic model in the freedom degree equation of mechanical model corresponding node, after being decoupled.
Preferably, numerical fitting uses least square method in S5 comprising following steps:
The wave force of discrete point form and fluctuation moment are expressed as at S51, multiple degrees of freedom spring
S52, polyfit function pair is used in MATLAB softwarePolynomial function linear fit is carried out, is obtained
To the wave function Δ F (t) of fitting.
Preferably, the method for the lumped mass kinetic model of each subsystem is established in the S6 the following steps are included:
S61, each subsystem kinetic model, i.e. tooth axle-wheel-bearing-box system are established according to generalized finite metatheory
Full Degree-of-freedom Coupling kinetic model, utilize formula (1) indicate tooth axle-wheel-bearing-box system mass motion differential side
Journey,
Wherein,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0Finger system external applied load vector；
S62, tooth axle-wheel-full Degree-of-freedom Coupling kinetic model of bearing-box system is established, specifically includes following step
It is rapid:
Input shaft and output shaft are divided into several shaft parts by S621, the kinetic model for establishing input shaft and output shaft,
The both ends of each shaft part are node, and the unit that two adjacent nodes are formed on same axis is shaft part unit, according to Timoshenko
The kinetic simulation of beam principle foundation shaft part unit as shown in formula (2)
Wherein,
Ms21=Ms12, Ms22=Ms11
MsFor the mass matrix of shaft part unit；
CsFor the damping matrix of shaft part unit；
KsFor the stiffness matrix of shaft part unit；
XsFor the displacement column vector of shaft part unit；
ρ is density of material (kg/m3)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
J is polar moment of inertia (m4)；
E is elasticity modulus of materials (Pa)；
G is material shearing elasticity model (Pa)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
IxFor the cross sectional moment of inertia (m in yz coordinate plane4)；
IyFor the cross sectional moment of inertia (m in xz coordinate plane4)；
J is polar moment of inertia (m4)；
K is correction factor (k=10/9)；
D is shaft part outer diameter (m)；
D is shaft part internal diameter (m)；
S622, the kinetic model for establishing driving gear and driven gear, gear are equivalent to a cylindrical body, cylinder outer diameter
Size is the reference diameter of gear, and inner diameter size is gear hub internal diameter, and meshing relationship table is shown to employ nibbles with time-varying
The spring for closing rigidity connects two wheel bodys, and the rotary inertia I of gear cylinder body is calculated using formula (3)s,
Wherein,
D is outer diameter (m)；
D is internal diameter (m)；
L is the length (m) of equivalent cylindrical；
ρ is density of material (kg/m3)；
The rotary inertia of gear cylinder body is added at the axis corresponding node that gear is connected with axis, the time-varying of gear engages
Rigidity is according to ISO-6366 and instantaneous contact line length computation；
S623, establish bearing unit kinetic model, ignore the quality and inertia of bearing, be reduced to rigidity and
The spring of damping, rigidity value and damping value are measured according to experiment；
S624, the kinetic model for establishing box unit are calculated gear case body using subsctructure method, in cabinet bearing
Host node is arranged in hole center, extracts cabinet and condenses upon the effective stiffness matrix K at host nodegWith mass matrix Mg, utilize formula (4)
The kinetic model of cabinet is established,
Wherein,
Cg=α0Mg+α1Kg
MgFor the mass matrix of box unit；
CgFor the damping matrix of box unit；
KgFor the stiffness matrix of box unit；
XgFor the displacement column vector of box unit；
α0It is mass ratio coefficient in Rayleigh damping；
α1It is rigidity proportionality coefficient in Rayleigh damping；
S625, according to the interconnected relationship of gear train inner body, assemble each mass matrix, stiffness matrix and damping
Matrix, driving gear and driven gear are reduced to a node on coupled axis, this node merges with node on primitive axis,
Inertia is superimposed, and the node of installation site on axis is connected with bearing for bearing unit one end, at one end and box unit bearing hole
Node is connected, and according to the corresponding relationship between node, the submatrix of different units type is added to the corresponding position of global matrix
It sets, forms the kinetic model of system；
S63, the freedom degree that wave function at obtained coupler spring node is added to kinetics equation corresponding node
Place, obtains the mass motion differential equation of the subsystem as shown in following formula (5)
Wherein,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0Finger system external applied load vector；
Δ F (t) refers to wave function column vector at coupler spring node.
Preferably, the damping matrix C in the step S621sIt is determined using the Rayleigh damping form of formula (6),
Cm=α0Mm+α1Km(m=1,2 ...) (6)
In formula:
CmRefer to the damped coefficient of m first order mode；
α0It is mass ratio coefficient in Rayleigh damping；
α1It is rigidity proportionality coefficient in Rayleigh damping；
MmIt is the vibration shape quality of m first order mode；
KmIt is the rigidity of m first order mode；
M is the order of the vibration shape.
Compared with prior art, a kind of multiple-input and multiple-output gear train assembly dynamic modeling method provided by the invention
Beneficial effect be:
1, the present invention establishes the multiple-input and multiple-output gear train assembly kinetic model after decoupling, improves system dynamic
The computational efficiency of response is learned, and can guarantee certain computational accuracy；
2, the present invention is based on the model visualization degree height that many-body dynamics method obtains, it is easy to modify；
3, the present invention can reduce the operation difficulty of researcher, save manpower；
4, practicability of the present invention is good, is worthy to be popularized.
Detailed description of the invention
Fig. 1 is a kind of multiple-input and multiple-output gear train assembly dynamic modeling method flow chart；
Fig. 2 is dual input list output transmission systems configuration schematic diagram；
Fig. 3 is G1 (G2) subsystem multi-rigid model schematic diagram；
Fig. 4 is G3 subsystem multi-rigid model schematic diagram；
Fig. 5 is G1 system dynamics model schematic diagram.
Specific embodiment
The present invention provides a kind of multiple-input and multiple-output gear train assembly dynamic modeling methods, below with reference to Fig. 1's
Flow diagram, the present invention will be described.
As shown in Figure 1, the technical scheme is that
A kind of multiple-input and multiple-output gear train assembly dynamic modeling method, comprising the following steps:
S1, multiple-input and multiple-output gear train assembly is divided by multiple sub-drivers from shaft coupling according to drive connection
System；
S2, rigid body modeling is carried out to the composition components in each sub-driver system respectively, is assembled according to assembly relation
Form subsystem model；
S3, shaft coupling is reduced to the multiple degrees of freedom spring with rigidity and damping, passes through connection between each subsystem model
Axis device is connected, and constitutes the overall model of multiple-input and multiple-output gear train assembly；
S4, primary condition is applied to the overall model of multiple-input and multiple-output gear train assembly in a simulated environment, to it
Simulation Analysis on Multi-body Dynamics is carried out, the discrete data of wave force and fluctuation moment at multiple degrees of freedom spring is obtained；
S5, numerical fitting, the wave function changed over time are carried out to the discrete data of wave force and fluctuation moment
ΔF(t)；
It is dynamic to be added to lumped mass by S6, the lumped mass kinetic model for establishing each subsystem by wave function Δ F (t)
Each subsystem kinetic model in the freedom degree equation of mechanical model corresponding node, after being decoupled.
Further, numerical fitting uses least square method in the S5 comprising following steps:
The wave force of discrete point form and fluctuation moment are expressed as at S51, multiple degrees of freedom spring
S52, polyfit function pair is used in MATLAB softwarePolynomial function linear fit is carried out, is obtained
To the wave function Δ F (t) of fitting.
Further, the method that the lumped mass kinetic model of each subsystem is established in the S6 includes following step
It is rapid:
S61, each subsystem kinetic model, i.e. tooth axle-wheel-bearing-box system are established according to generalized finite metatheory
Full Degree-of-freedom Coupling kinetic model, utilize formula (1) indicate tooth axle-wheel-bearing-box system mass motion differential side
Journey,
Wherein,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0Finger system external applied load vector；
S62, tooth axle-wheel-full Degree-of-freedom Coupling kinetic model of bearing-box system is established, specifically includes following step
It is rapid:
Input shaft and output shaft are divided into several shaft parts by S621, the kinetic model for establishing input shaft and output shaft,
The both ends of each shaft part are node, and the unit that two adjacent nodes are formed on same axis is shaft part unit, according to Timoshenko
The kinetic model of beam principle foundation shaft part unit as shown in formula (2)
Wherein,
Ms21=Ms12, Ms22=Ms11
MsFor the mass matrix of shaft part unit；
CsFor the damping matrix of shaft part unit；
KsFor the stiffness matrix of shaft part unit；
XsFor the displacement column vector of shaft part unit；
ρ is density of material (kg/m3)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
J is polar moment of inertia (m4)；
E is elasticity modulus of materials (Pa)；
G is material shearing elasticity model (Pa)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
IxFor the cross sectional moment of inertia (m in yz coordinate plane4)；
IyFor the cross sectional moment of inertia (m in xz coordinate plane4)；
J is polar moment of inertia (m4)；
K is correction factor (k=10/9)；
D is shaft part outer diameter (m)；
D is shaft part internal diameter (m)；
S622, the kinetic model for establishing driving gear and driven gear, gear are equivalent to a cylindrical body, cylinder outer diameter
Size is the reference diameter of gear, and inner diameter size is gear hub internal diameter, and meshing relationship table is shown to employ nibbles with time-varying
The spring for closing rigidity connects two wheel bodys, and the rotary inertia I of gear cylinder body is calculated using formula (3)s,
Wherein,
D is outer diameter (m)；
D is internal diameter (m)；
L is the length (m) of equivalent cylindrical；
ρ is density of material (kg/m3)；
The rotary inertia of gear cylinder body is added at the axis corresponding node that gear is connected with axis, the time-varying of gear engages
Rigidity is according to ISO-6366 and instantaneous contact line length computation；
S623, establish bearing unit kinetic model, ignore the quality and inertia of bearing, be reduced to rigidity and
The spring of damping, rigidity value and damping value are measured according to experiment；
S624, the kinetic model for establishing box unit are calculated gear case body using subsctructure method, in cabinet bearing
Host node is arranged in hole center, extracts cabinet and condenses upon the effective stiffness matrix K at host nodegWith mass matrix Mg, utilize formula (4)
Establish the kinetic model of cabinet
Wherein,
Cg=α0Mg+α1Kg
MgFor the mass matrix of box unit；
CgFor the damping matrix of box unit；
KgFor the stiffness matrix of box unit；
XgFor the displacement column vector of box unit；
α0It is mass ratio coefficient in Rayleigh damping；
α1It is rigidity proportionality coefficient in Rayleigh damping；
S625, according to the interconnected relationship of gear train inner body, assemble each mass matrix, stiffness matrix and damping
Matrix, driving gear and driven gear are reduced to a node on coupled axis, this node merges with node on primitive axis,
Inertia is superimposed, and the node of installation site on axis is connected with bearing for bearing unit one end, at one end and box unit bearing hole
Node is connected, and according to the corresponding relationship between node, the submatrix of different units type is added to the corresponding position of global matrix
It sets, forms the kinetic model of system；
S63, the freedom degree that wave function at obtained coupler spring node is added to kinetics equation corresponding node
Place, obtains the mass motion differential equation of the subsystem as shown in following formula (5)
Wherein,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0Finger system external applied load vector；
Δ F (t) refers to wave function column vector at coupler spring node.
Further, the damping matrix C in the step S621sIt is determined using the Rayleigh damping form of formula (6),
Cm=α0Mm+α1Km(m=1,2 ...) (6)
In formula:
CmRefer to the damped coefficient of m first order mode；
α0It is mass ratio coefficient in Rayleigh damping；
α1It is rigidity proportionality coefficient in Rayleigh damping；
MmIt is the vibration shape quality of m first order mode；
KmIt is the rigidity of m first order mode；
M is the order of the vibration shape.
Embodiment 1
The present embodiment introduces a specific embodiment of the invention by taking the gear train assembly that dual input list exports as an example, main
Want the following steps are included:
S1, multiple-input and multiple-output gear train assembly is divided by multiple sub-drivers from shaft coupling according to drive connection
System；
S2, rigid body modeling is carried out to the composition components in each sub-driver system respectively, is assembled according to assembly relation
Form subsystem model；
S3, shaft coupling is reduced to the multiple degrees of freedom spring with rigidity and damping, passes through connection between each subsystem model
Axis device is connected, and constitutes the overall model of multiple-input and multiple-output gear train assembly；
S4, primary condition is applied to the overall model of multiple-input and multiple-output gear train assembly in a simulated environment, to it
Simulation Analysis on Multi-body Dynamics is carried out, the discrete data of wave force and fluctuation moment at multiple degrees of freedom spring is obtained；
S5, numerical fitting, the wave function changed over time are carried out to the discrete data of wave force and fluctuation moment
ΔF(t)；
It is dynamic to be added to lumped mass by S6, the lumped mass kinetic model for establishing each subsystem by wave function Δ F (t)
Each subsystem kinetic model in the freedom degree equation of mechanical model corresponding node, after being decoupled.
The step S1 specifically includes the following steps:
In conjunction with the embodiments, as shown in Fig. 2, dual input list output gear transmission system transmission system is by three gearbox drives
System group is connected by shaft coupling and is formed, therefore overall gear train assembly is divided into three sub- transmission systems: G1 subsystem,
G2 subsystem, G3 subsystem.
The step S2 specifically includes the following steps:
Each sub-driver system components mainly include gear, input shaft, output shaft, bearing and cabinet, according to known to component
After parameter and size respectively model gear, input shaft, input shaft, cabinet, shape is assembled further according to assembly relation
At overall subsystem.
It is illustrated in figure 3 G1 subsystem multi-rigid model schematic diagram, composed structure includes G1 gear housing 1, G1 driven
Gear 2, G1 output shaft 3, G1 input shaft 4 and G1 driving gear 5.
Wherein G1 driving gear 5 is fixedly connected with G1 input shaft 4, and G1 driven gear 2 is fixedly connected with G1 output shaft 3, G1
With the spring connection with periodical time-varying rigidity, G1 input shaft 4, G1 output shaft 3 between driving gear 5 and G1 driven gear 2
It is connected with G1 gear housing 1 by the multiple degrees of freedom spring that bearing simplifies, spring one end is connected to the bearing central point of hole of cabinet
Place, one end are connected to bearing at the installation site center on axis, repeat above step, obtain the multi-rigid body power of G1 subsystem
Learn model.
The Multi-body dynamic model of G2 subsystem is as the Multi-body dynamic model structure of G1 subsystem.
It is illustrated in figure 4 G3 subsystem multi-rigid model schematic diagram, composed structure includes G3 gear housing 6, G3 active
Gear 7, G3 input shaft 8, G3 jackshaft 9, G3 intermediate reduction gear gear 10, G3 output shaft 11, G3 driven gear 12.
Wherein, G3 driving gear 7 is fixedly connected with G3 input shaft 8, G3 intermediate reduction gear gear 10 and the fixed company of G3 jackshaft 9
It connects, G3 driven gear 12 is fixedly connected with G3 output shaft 11, G3 driving gear 7 and G3 intermediate reduction gear gear 10, G3 intermediate reduction gear
With the spring connection with periodical time-varying rigidity, G3 input shaft 8, G3 jackshaft 9, G3 between gear 10 and G3 driven gear 12
Output shaft 11 is connected with G3 gear housing 6 by the multiple degrees of freedom spring that bearing simplifies, and spring one end is connected to the bearing of cabinet
At central point of hole, one end is connected to bearing at the installation site center on axis, so far obtains the multi-rigid body power of G3 subsystem
Learn model.
The step S3 specifically includes the following steps:
The quality of shaft coupling and rotary inertia are divided equally, are added on coupled shaft part；Shaft coupling is simplified later
For the multiple degrees of freedom spring with rigidity and damping, mathematical expression form are as follows:
In formula:
Fx、Fy、FzRespectively represent the component of shaft coupling transmission force in the x, y, z-directions；
Tα、Tβ、TγShaft coupling transmitting torque is respectively represented in the moment of components around x, y, z direction；
X, Y, Z, α, β, γ indicate shaft coupling displacement and angular displacement；
Indicate the speed and angular speed of shaft coupling；
Ki(i=x, xy ...) indicates shaft coupling items rigidity value；
Ci(i=x, xy ...) indicates shaft coupling items damping value；
Fx1、Fy1、Fz1Respectively represent the component of shaft coupling nominal force in the x, y, z-directions；
Tα、Tβ、TγShaft coupling name torque is respectively represented in the moment of components around x, y, z direction.
A coupler spring, G2 output shaft are coupled between the output shaft node of G1 system and the input shaft node of G3 system
A coupler spring is coupled between node and G3 input shaft node.
Further, node selection principle are as follows: selection system actual configuration in shaft coupling respectively with gearbox drive axis coupling
The central point of sympodium section, at this point, coupling is formed overall dual input list by each subsystem exports rigid model.
The step S4 specifically includes the following steps:
In conjunction with the actual condition that dual input list exports, corresponding revolving speed is applied respectively to the input shaft of G1 system and G2 system
And power, apply load torque in the output shaft of G3 system, the revolving speed of output end is calculated according to following gear drive expression formula:
In formula:
noutTo export revolving speed；
ninFor input speed；
z1…ziFor the number of teeth of driving gear；
z2…znFor the number of teeth of driven gear.
The occurrence of the stiffness matrix and damping matrix of bearing and shaft coupling is measured by experiment；
The mesh stiffness of gear obtains time-variant mesh stiffness according to ISO-6366 and instant contact line length, by obtain when
Become mesh stiffness to be added at the gear engaging element in multiple-input and multiple-output gear train assembly overall model.
Multi-body Dynamic Analysis is carried out to established dual input list output gear transmission system, obtains coupler spring section
Wave force and fluctuation moment at point.
In the step S5 numerical fitting use least square method specifically includes the following steps:
The wave force of discrete point form and fluctuation moment are expressed as at S51, multiple degrees of freedom spring
S52, polyfit function pair is used in MATLAB softwarePolynomial function linear fit is carried out, is obtained
To the wave function Δ F (t) of fitting.
The step S6 specifically includes the following steps:
S61, subsystems kinetic model, i.e. tooth axle-wheel-bearing-cabinet system are established according to generalized finite metatheory
The full Degree-of-freedom Coupling kinetic model of system.It is by taking G1 subsystem as an example, subsystem is discrete for different types of list shown in Fig. 5
Member and node, cell type specifically include carriage spring unit 13, driven gear unit 14, joint unit 15, gear housing
Unit 16, driving gear unit 17, shaft part unit 18, engaging element 19, coupler spring unit 20.
Wherein, driving gear unit 17 is fixedly connected with shaft part unit 18 by connecting node, driven gear unit 14 with
Shaft part unit 18 is fixedly connected by connecting node, and engaging element 19 is used between driving gear unit 17 and driven gear unit 14
Connection, joint unit 15 are connect with shaft part unit 18 by coupler spring unit 20, shaft part unit 18 and gear housing list
Member 16 is connected by carriage spring unit 13, and spring one end is connected at the bearing hole node of cabinet, and one end is connected to bearing and exists
At installation site node on axis, above step is repeated, the Multi-body dynamic model of G1 subsystem is obtained.
The Multi-body dynamic model of G2 subsystem is as the Multi-body dynamic model structure of G1 subsystem.
Force analysis then is carried out to each unit, is established to corresponding differential equation of motion, further according to FInite Element to each
The quality of unit, rigidity, damping matrix are assembled, obtain as follows be using each unit modal displacement as generalized coordinates
Entirety of the uniting differential equation
In above formula,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0Finger system external applied load vector.
S62, it establishes tooth axle-wheel-full Degree-of-freedom Coupling kinetic model of bearing-box system and specifically includes following step
It is rapid:
Input shaft and output shaft are divided into several shaft parts by S621, the kinetic model for establishing input shaft and output shaft,
The both ends of each shaft part are node, and the unit that two adjacent nodes are formed on same axis is shaft part unit, according to Timoshenko
Beam principle establishes the kinetic model of shaft part unit as follows
Wherein,
Ms21=Ms12, Ms22=Ms11
MsFor the mass matrix of shaft part unit；
CsFor the damping matrix of shaft part unit；
KsFor the stiffness matrix of shaft part unit；
XsFor the displacement column vector of shaft part unit；
ρ is density of material (kg/m3)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
J is polar moment of inertia (m4)；
E is elasticity modulus of materials (Pa)；
G is material shearing elasticity model (Pa)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
IxFor the cross sectional moment of inertia (m in yz coordinate plane4)；
IyFor the cross sectional moment of inertia (m in xz coordinate plane4)；
J is polar moment of inertia (m4)；
K is correction factor (k=10/9)；
D is shaft part outer diameter (m)；
D is shaft part internal diameter (m).
S622, the kinetic model for establishing driving gear and driven gear, gear are equivalent to a cylindrical body, cylinder outer diameter
Size is the reference diameter of gear, and inner diameter size is gear hub internal diameter, and meshing relationship table is shown to employ nibbles with time-varying
The spring for closing rigidity connects two wheel bodys, and the rotary inertia I of gear cylinder body is calculated using following formulas,
Wherein,
D is outer diameter (m)；
D is internal diameter (m)；
L is the length (m) of equivalent cylindrical；
ρ is density of material (kg/m3)；
The rotary inertia of gear cylinder body is added at the axis corresponding node that gear is connected with axis, the time-varying of gear engages
Rigidity is according to ISO-6366 and instantaneous contact line length computation.
S623, establish bearing unit kinetic model, ignore the quality and inertia of bearing, be reduced to rigidity and
The multiple degrees of freedom spring of damping, rigidity value and damping value are measured according to experiment, mathematical expression form are as follows:
In formula:
Fx、Fy、FzRespectively represent the component of bearing in the x, y, z-directions；
Tα、Tβ、TγBearing moment is respectively represented in the moment of components around x, y, z direction；
X, Y, Z, α, β, γ indicate bearing different directions displacement and angular displacement；
Indicate the speed and angular speed of the different directions of bearing；
Ki(i=x, xy ...) indicates bearing items rigidity value；
Ci(i=x, xy ...) indicates bearing items damping value.
S624, the kinetic model for establishing box unit are calculated gear case body using subsctructure method, in cabinet bearing
Host node is arranged in hole center, extracts cabinet and condenses upon the effective stiffness matrix K at host nodegWith mass matrix Mg, built using following formula
The kinetic model of vertical cabinet,
Wherein,
C)=α0Mg+α1Kg
MgFor the mass matrix of box unit；
CgFor the damping matrix of box unit；
KgFor the stiffness matrix of box unit；
XgFor the displacement column vector of box unit；
α0It is mass ratio coefficient in Rayleigh damping；
α1It is rigidity proportionality coefficient in Rayleigh damping；
S625, according to the interconnected relationship of gear train inner body, assemble each mass matrix, stiffness matrix and damping
Matrix, driving gear and driven gear are reduced to a node on coupled axis, this node merges with node on primitive axis,
Inertia is superimposed, and the node of installation site on axis is connected with bearing for bearing unit one end, at one end and box unit bearing hole
Node is connected, and according to the corresponding relationship between node, the submatrix of different units type is added to the corresponding position of global matrix
It sets, forms the kinetic model of system；
S63, the freedom degree that wave function at obtained coupler spring node is added to kinetics equation corresponding node
Place, obtains the mass motion differential equation of subsystem as follows
Wherein,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0Finger system external applied load vector；
Δ F (t) refers to wave function column vector at coupler spring node.
The present invention establishes the multiple-input and multiple-output gear train assembly kinetic model after decoupling, improves system dynamics
The computational efficiency of response, and can guarantee certain computational accuracy, meanwhile, the model visualization obtained based on many-body dynamics method
Degree is high, is easy to modify, and can reduce researcher's operation difficulty, saves manpower, practicability is good, is worthy to be popularized.
Disclosed above is only preferable specific embodiment of the invention, and still, the embodiment of the present invention is not limited to this,
What anyone skilled in the art can be thought variation should all fall into protection scope of the present invention.
Claims (4)
1. a kind of multiple-input and multiple-output gear train assembly dynamic modeling method, which comprises the following steps:
S1, multiple-input and multiple-output gear train assembly is divided by multiple sub-drivers systems from shaft coupling according to drive connection；
S2, rigid body modeling is carried out to the composition components in each sub-driver system respectively, formation is assembled according to assembly relation
Subsystem model；
S3, shaft coupling is reduced to the multiple degrees of freedom spring with rigidity and damping, passes through shaft coupling between each subsystem model
It is connected, constitutes the overall model of multiple-input and multiple-output gear train assembly；
S4, primary condition is applied to the overall model of multiple-input and multiple-output gear train assembly in a simulated environment, it is carried out
Simulation Analysis on Multi-body Dynamics obtains the discrete data of wave force and fluctuation moment at multiple degrees of freedom spring；
S5, numerical fitting, the wave function Δ F changed over time are carried out to the discrete data of wave force and fluctuation moment
(t)；
Wave function Δ F (t) is added to lumped mass dynamics by S6, the lumped mass kinetic model for establishing each subsystem
Each subsystem kinetic model in the freedom degree equation of model corresponding node, after being decoupled.
2. a kind of multiple-input and multiple-output gear train assembly dynamic modeling method according to claim 1, feature exist
In numerical fitting uses least square method in the S5 comprising following steps:
The wave force of discrete point form and fluctuation moment are expressed as at S51, multiple degrees of freedom spring
S52, polyfit function pair is used in MATLAB softwarePolynomial function linear fit is carried out, is intended
The wave function Δ F (t) of conjunction.
3. a kind of multiple-input and multiple-output gear train assembly dynamic modeling method according to claim 1, feature exist
In, the method for the lumped mass kinetic model of each subsystem is established in the S6 the following steps are included:
S61, each subsystem kinetic model is established according to generalized finite metatheory, i.e. tooth axle-wheel-bearing-box system is complete
Degree-of-freedom Coupling kinetic model indicates tooth axle-wheel-bearing-box system mass motion differential equation using formula (1),
Wherein,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0Finger system external applied load vector；
S62, tooth axle-wheel-full Degree-of-freedom Coupling kinetic model of bearing-box system is established, specifically includes the following steps:
Input shaft and output shaft are divided into several shaft parts, each by S621, the kinetic model for establishing input shaft and output shaft
The both ends of shaft part are node, and the unit that two adjacent nodes are formed on same axis is shaft part unit, former according to Timoshenko beam
Reason establishes the kinetic simulation of the shaft part unit as shown in formula (2)
Wherein,
Ms21=Ms12, Ms22=Ms11
MsFor the mass matrix of shaft part unit；
CsFor the damping matrix of shaft part unit；
KsFor the stiffness matrix of shaft part unit；
XsFor the displacement column vector of shaft part unit；
ρ is density of material (kg/m3)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
J is polar moment of inertia (m4)；
E is elasticity modulus of materials (Pa)；
G is material shearing elasticity model (Pa)；
A is the cross-sectional area (m of unit2)；
L is the length (m) of unit；
IxFor the cross sectional moment of inertia (m in yz coordinate plane4)；
IyFor the cross sectional moment of inertia (m in xz coordinate plane4)；
J is polar moment of inertia (m4)；
K is correction factor (k=10/9)；
D is shaft part outer diameter (m)；
D is shaft part internal diameter (m)；
S622, the kinetic model for establishing driving gear and driven gear, gear are equivalent to a cylindrical body, cylinder outer diameter size
For the reference diameter of gear, inner diameter size is gear hub internal diameter, and meshing relationship table is shown to employ rigid with time-varying engagement
The spring of degree connects two wheel bodys, and the rotary inertia I of gear cylinder body is calculated using formula (3)s,
Wherein,
D is outer diameter (m)；
D is internal diameter (m)；
L is the length (m) of equivalent cylindrical；
ρ is density of material (kg/m3)；
The rotary inertia of gear cylinder body is added at the axis corresponding node that gear is connected with axis, the time-variant mesh stiffness of gear
According to ISO-6366 and instantaneous contact line length computation；
S623, bearing unit kinetic model is established, ignores the quality and inertia of bearing, is reduced to rigidity and damping
Spring, rigidity value and damping value are measured according to experiment；
S624, the kinetic model for establishing box unit are calculated gear case body using subsctructure method, in cabinet bearing hole
Host node is arranged in the heart, extracts cabinet and condenses upon the effective stiffness matrix K at host nodegWith mass matrix Mg, established using formula (4)
The kinetic model of cabinet,
Wherein,
Cg=α0Mg+α1Kg
MgFor the mass matrix of box unit；
CgFor the damping matrix of box unit；
KgFor the stiffness matrix of box unit；
XgFor the displacement column vector of box unit；
α0It is mass ratio coefficient in Rayleigh damping；
α1It is rigidity proportionality coefficient in Rayleigh damping；
S625, according to the interconnected relationship of gear train inner body, assemble each mass matrix, stiffness matrix and damping square
Battle array, driving gear and driven gear are reduced to a node on coupled axis, this node merges with node on primitive axis, are used to
Measure superimposed, the node of installation site on axis is connected with bearing for bearing unit one end, saves at one end and box unit bearing hole
Point is connected, and according to the corresponding relationship between node, the submatrix of different units type is added to the corresponding position of global matrix,
The kinetic model of formation system；
S63, wave function at obtained coupler spring node is added at the freedom degree of kinetics equation corresponding node, is obtained
To the mass motion differential equation of the subsystem as shown in following formula (5)
Wherein,
M refers to system total quality matrix；
C refers to system integral damping matrix；
K (t) refers to system Bulk stiffness matrix；
X (t) refers to all modal displacement column vectors；
P0 refers to system external applied load vector；
Δ F (t) refers to wave function column vector at coupler spring node.
4. a kind of multiple-input and multiple-output gear train assembly dynamic modeling method according to claim 3, feature exist
In damping matrix C in the step S621sIt is determined using the Rayleigh damping form of formula (6),
Cm=α0Mm+α1Km(m=1,2...) (6)
In formula:
CmRefer to the damped coefficient of m first order mode；
α0It is mass ratio coefficient in Rayleigh damping；
α1It is rigidity proportionality coefficient in Rayleigh damping；
MmIt is the vibration shape quality of m first order mode；
KmIt is the rigidity of m first order mode；
M is the order of the vibration shape.
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| CN112417609A (en) * | 2020-12-15 | 2021-02-26 | 中国第一汽车股份有限公司 | Steering transmission shaft optimization design method, computer equipment and storage medium |
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| CN114670205B (en) * | 2022-04-29 | 2023-11-21 | 长沙长泰机器人有限公司 | Mechanical transmission decoupling method for multi-axis robot |
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