CN107220421A - A kind of spatial complex flexible structure dynamics of multibody systems modeling and computational methods - Google Patents
A kind of spatial complex flexible structure dynamics of multibody systems modeling and computational methods Download PDFInfo
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Abstract
The present invention provides a kind of spatial complex flexible structure dynamics of multibody systems modeling and computational methods, and detailed process is：Step 101：Build threedimensional model and assign parameter；Step 102：Mesh generation is carried out to model；Step 103：Calculate grid cell generalized coordinates vector q and generalized velocity vectorStep 104：Set up the constraint equation of grid model；Step 105：According to grid cell type, generalized coordinates vector, generalized velocity vector sum constraint equation, Equations of Motion of Multibody Systems is set up based on lagrange equations of the first kind；Step 106：The Equations of Motion of Multibody Systems built is solved；So far modeling and solution to spatial complex flexible structure dynamics of multibody systems problem are completed.The present invention provides a kind of simple and effective dynamic modeling method for flexible structure multi-body system, improves computational efficiency.
Description
Technical field
The invention belongs to space structure dynamics calculation technical field, and in particular to a kind of many bodies of spatial complex flexible structure
System dynamic modeling and computational methods.
Background technology
In recent years, the development of China's Space Science and Technology is defended in the urgent need to grasping the in-orbit expansion technique of flexible space structure with meeting
The great demands such as star communication, space-based earth observation and survey of deep space.This kind of space structure it is general main by rigid rod, flexible beam,
The components such as plate, shell and flexible rope are constituted, between each component in addition to the position constraint at hinged joint, also there are a large amount of synchronous, slidings
Deng nonlinear restriction.This kind of space structure expansion yardstick is big simultaneously, flexible member is more, configuration is complicated, and its expansion process can be presented
Non-linear coupling dynamics between system grand movement and flexible member large deformation.The states such as the U.S., Russia are in its space flight
In task, once occurred a lot of structure expansion failures.Because the expansion dynamics ground experiment difficulty of this class formation is big, can not be complete
Offsetting influence of gravity, therefore its dynamic numerical simulation for deploying process is to ensure that structure deploys successful key technology.
For such spatial complex flexible structure multi-body system, the Research Thinking of classical dynamics of multibody systems method is:
Adhere to floating coordinate system on flexible structure, beformable body grand movement described relative to the motion of absolute coordinate system with it,
Pass through the deformation of mode polycondensation approximate description flexible structure Relative Floating coordinate system.Such method be typically led to it is small rotate, it is small
Deformation is it is assumed that the elastic deformation of flexible structure is separately considered with rigid body displacement, progress decoupling processing.Usual this method is only
Large deformation, the big displacement problem of flexible structure can be handled by substantial increase element number, so as to cause that calculation scale is big, count
Calculate efficiency low.In addition, the mechanical essence characteristic that this method can not accurately reflect during the expansion of spatial complex flexible structure.
Because there is the load-up condition of many nonlinear restrictions, pair clearance and complexity, mesh in spatial complex flexible structure multi-body system
There are many difficulties in terms of system dynamic modeling, numerical solution efficiency in preceding business software.
The content of the invention
In view of this, the invention provides a kind of modeling of spatial complex flexible structure dynamics of multibody systems and calculating side
Method, provides a kind of simple and effective dynamic modeling method for flexible structure multi-body system, improves computational efficiency.
Realize that technical scheme is as follows：
Step 101：Build threedimensional model and assign parameter：According to the material object of required solution, the threedimensional model in kind is set up,
Input material parameter, geometric parameter and the space structure topological relation of institute's established model；
Step 102：Mesh generation：The threedimensional model after parameter is assigned based on step 101 and carries out mesh generation, using absolute
Node coordinate method, geometry Precise equation wait geometric analysis method to determine grid cell type, and the threedimensional model after division is defined
For grid model；
Step 103：The grid divided according to step 102 determines start node position and the start node of each grid cell
Speed, based on unit start node position and start node speed calculate grid cell generalized coordinates vector q and generalized velocity to
Amount
Step 104：The generalized coordinates vector sum generalized velocity vector obtained according to step 103, sets the outer of grid model
Portion's load and boundary condition, the boundary condition based on grid model set up the constraint equation of grid model, and when setting emulation total
Long T and simulation time step-length h；
Step 105：According to the grid cell type of step 102-104 acquisitions, generalized coordinates vector, generalized velocity vector sum
Constraint equation, Equations of Motion of Multibody Systems is set up based on lagrange equations of the first kind, as follows：
Wherein M is mass of system battle array,For generalized acceleration vector, F () is the elastic force vector of system, and Q () is to be
The generalized external force vector of system, Φ () and Φq() is respectively the constraint equation vector sum constraint equation vector that step 104 is obtained
To the partial derivative matrix of generalized coordinates vector, ()qFor the partial derivative of function pair generalized coordinates vector, λ is Lagrange multiplier
Vector, t is simulation time, and t ∈ [0, T], T is matrix transposition；
Step 106：Time domain discrete is carried out to the Equations of Motion of Multibody Systems built, many body system after time domain discrete is obtained
System kinetics equation, is iterated to the Equations of Motion of Multibody Systems after discrete, and the linear side to being generated in iterative process
Journey group carries out Parallel implementation, output descriptor coordinate vector q and Lagrange multiplier vector λ, so far completes soft to spatial complex
The modeling and solution of property structure dynamics of multibody systems problem.
Further, the step 106 is specially：
Step 601：Equations of Motion of Multibody Systems is obtained according to step 105, Equations of Motion of Multibody Systems is carried out from
Dissipate, discrete rear Equations of Motion of Multibody Systems is shown in formula (2)；
Step 602：Equations of Motion of Multibody Systems after discrete is carried out solving the generalized coordinates for obtaining the (n+1)th step vector
qn+1, generalized velocity vectorWith Lagrange multiplier vector λn+1；
Step 603：Time t is judged according to step 602n+1Whether total simulation time T is less than, if current time tn+1It is not less than
Total simulation time T, then calculate and stop；Otherwise, n adds 1 automatically, repeats step 601-603, calculates generalized coordinates vector qn+1、
Generalized velocity vectorMake time tn+1=tn+ h, until tn+1More than or equal to T, output descriptor coordinate vector qn+1And draw
Ge Lang multipliers vector λn+1, wherein qn+1For q, λn+1For λ.
Further, discrete adopt with the following method is carried out to Equations of Motion of Multibody Systems：
Equations of Motion of Multibody Systems is carried out using broad sense-alpha implicit time integrations algorithm following discrete：
Wherein
In formula (4), β and γ take γ >=1/2 and β >=(1/2+ γ) to determine the algorithm parameter of computational accuracy and efficiency2/
4.N is iterations, and h is simulation time step-length, for dissipative system high frequency response, introduces new algorithm vector parameters a, wherein
Vector parameters a meets relation：
The choosing method of each parameter is as follows in formula (4) and formula (5):
WhereinFor the spectral radius of algorithm,αmAnd αfIt is that broad sense-alpha implicit time integrations are calculated
Method parameter.
Further, q is represented with q in this stepn+1, λ represents λn+1, the step 602 further comprises：
Step 6021：Solving precision Tol is set；
Step 6022：Using formula (2) described in Newton-Rapson iteratives, following linear algebraic equation systems are obtained：
Wherein Δ q and Δ λ are the increments of current time t system generalized coordinates vector sum Lagrange multiplier vector, wherein
WithTo meet the algorithm parameter of following relation:
Wherein,WithIt is intermediate variable, ψ and Γ are intermediary matrix,For the inclined of function pair generalized velocity vector
Derivative；
Step 6023：The system of linear equations obtained according to step 6021 calculates current residue vector field homoemorphism r；
Wherein, | | | | it is mould；
Step 6024：Current generalized coordinates vector is updated to the solution of system of linear equations according to step 6022 and glug is bright
Day multiplier vector；
Step 6025：Judge whether iteration result restrains according to the residual vector mould that step 6023 is obtained, if r>Tol, then
The generalized coordinates vector and Lagrange multiplier vector obtained is updated according to step 6024, step 6022-6025 is repeated,
Until r<Tol or r=Tol, generalized coordinates vector q and Lagrange multiplier vector λ that output step 6024 is obtained.
Beneficial effect：
1) present invention is using absolute node coordinate method, the accurate beam theory of geometry and waits the finite element methods such as geometrical analysis
It is discrete to the progress of spatial complex flexible structure, motion and the Coupling Deformation problem of flexible structure have been taken into full account, has been spatial complex
The Dynamic Modeling of flexible structure multi-body system provides means.
2) present invention uses Region Decomposition and parallel computing, and spatial complex flexible structure many body system is greatly improved
System dynamics simulation efficiency.
Brief description of the drawings
Fig. 1 is to being modeled and computational methods flow chart containing complicated flexible multibody dynamics problem.
Fig. 2 is certain truss rope net antenna structure view.
The optional cell type schematic diagram in part that Fig. 3 is developed for the present invention.
Fig. 4 is bearing bush model structure schematic diagram.
Fig. 5 is synchromesh gear model structure schematic diagram.
Fig. 6 is prismatic pair model structure schematic diagram.
Fig. 7 is to system dynamics equation efficient numerical Integration Solving algorithm flow chart.
Fig. 8 is to system discrete nonlinear equation Efficient Solution algorithm flow chart.
Fig. 9 is that present invention modeling and derivation algorithm method are applied to certain truss rope net antenna, line number of going forward side by side value phantom antenna
Deploying portion moment structural representation.
Embodiment
The present invention will now be described in detail with reference to the accompanying drawings and examples.
For spatial complex flexible structure multi-body system, generally it is made up of numerous flexible members and flexible cable net, in expansion
During these components can produce very large deformation and large rotation, many volume modelings now assumed based on small deformation and small rotation
Analysis method can not meet engineering demand.Modeling for such flexible member mainly uses finite element method：As absolutely
To node coordinate method, geometry Precise equation and wait geometric analysis method etc..Wherein Shabana etc. is based on Continuum Mechanics and limited
First method proposes absolute node coordinate method (ANCF).Compared with classical dynamics of multibody systems method, this method is in inertial coordinate
Lower rigid motion and the elastic deformation that beformable body is described using unified interpolating function of system, is highly suitable for description while having big turn
The kinetics of deformable bodies of dynamic and large deformation.Developed in addition, the accurate beam theory of geometry is Simo etc. based on finite strain beam theory
A kind of nonlinear finite element method, the method can handle large deformation and big rotational problems efficiently and exactly, in recent years also by
It is widely used in the dynamics simulation of space deployable structure.For unified geometrical model and mechanics analysis model, both are realized
Seamless integration-, in finite element field, a kind of new numerical computation method based on spline theory of the proposition such as Hughes：
Deng geometrical analysis (IGA).
In summary, in the spatial complex flexible structure dynamics of multibody systems deployment analysis stage, how to consider and solve
Flexible member deform with the coupled problem of grand movement, and improve numerical simulation computational efficiency and develop urgently as Space Science and Technology
The technical problem of solution.The present invention proposes a kind of space structure dynamics of multibody systems modeling to the component containing large deformation and meter
Calculation method.
As shown in figure 1, it is to deploy dynamics to certain annular truss rope net antenna mechanism as shown in Figure 2 to implement typical case
Modeling and computational methods.
This method comprises the following steps：
Step 101：Build threedimensional model and assign parameter：According to the material object of required solution, the threedimensional model in kind is set up,
Input material parameter, geometric parameter and the space structure topological relation of institute's established model.Specifically include：According to annular truss rope net
The geometric parameter and space structure topological relation of antenna, set up annular truss rope net antenna threedimensional model in ProE softwares,
And input the material parameter of the different parts of annular truss antenna.Because wire reflects reticular density very little, and it is attached to forward pull
On the net, thus the influence of wire reflector net is ignored.
Step 102：Mesh generation：The threedimensional model after parameter is assigned based on step 101 and carries out mesh generation, using absolute
Node coordinate method, geometry Precise equation wait geometric analysis method to determine grid cell type, and the threedimensional model after division is defined
For grid model；According to model property and required precision, suitable cell type is chosen, as shown in figure 3, rigid model has：
The rigid unit described based on natural coordinates method, the reduction rigid unit described based on natural coordinates method, based on absolute section
The reference mode rigid unit of point coordinates method description；Beformable body unit has：The population parameter described based on absolute node coordinate method
Beam element, the reduction beam element described based on absolute node coordinate method, the slope based on the description of absolute node coordinate method are not
Continuous beam element, the Rectangular Thin Plate Element described based on absolute node coordinate and the triangle described based on absolute node coordinate
Shape thin-plate element etc., wherein absolute node coordinate method, geometry Precise equation wait geometric analysis method to be respectively provided with solution coupled characteristic
The characteristics of.
Step 103：Determine model generalized coordinates vector sum generalized velocity vector：The grid divided according to step 102 is determined
The start node position of each grid cell and start node speed, based on unit start node position and start node speedometer
Calculate grid cell generalized coordinates vector q and generalized velocity vectorCertain angle is arrived by the way that annular truss rope net antenna is drawn in,
So that it is determined that rope net and the initial generalized coordinates of truss and generalized velocity.
Step 104：The generalized coordinates vector sum generalized velocity vector obtained according to step 103, sets the outer of grid model
Portion's load and boundary condition, the boundary condition based on grid model set up the constraint equation of grid model, and when setting emulation total
Long T and simulation time step-length h.Annular truss rope net antenna emulation total duration is 500s, and simulation time step-length is set to 1e-
4s.Wherein external load is the pretension of rope net, is constrained to being hinged between rope net and truss, the kinematic pair between truss member
And it is affixed between truss and extending arm.The secondary modeling of three kinds of main movements is briefly described below：Rotary hinge constraint, synchronous gear
Wheel constraint and prismatic pair constraint.
Wherein to set up mode as follows for the constraint of joint rotary hinge：Rigid bearing bush model is set up to the joint of rotation,
As shown in Figure 4.To be fixedly connected between rigid bearing and bearing shell and the component being connected, then the rotation of connected links can lead to
The relative rotation of bearing bush is crossed to describe.Rotary hinge constraint equation between the bearing bush described using natural coordinates can table
It is shown as：
Wherein Φ1Constrained for rotary hinge, respectively by displacement constraint Φ1Φ is constrained with the anglec of rotation2Composition. WithRespectively global coordinate system lower bearing and bearing shell endpoint location coordinate.e1And e2It is local coordinate system lower bearing and bearing respectively
Unit vector, θ be bearing around bearing shell rotational angle, wherein
Wherein LbAnd LjIt is the length of bearing and bearing shell, u respectivelybAnd vbIt is the unit vector positioned at bearing shell end points b, ujAnd vj
It is the unit vector positioned at bearing end points j.
It is as follows that mode is set up in synchromesh gear constraint：
As shown in figure 5, synchromesh gear constraint ensures that the expanded angle in two neighboring face is consistent, i.e., first across cross bar (AB
Bar) angle between montant and second across the angle of cross bar (CD bars) between montant it is equal, its specific constraint equation
See below formula:
Wherein Φ2Constrained for synchromesh gear,WithIt is the axis direction vector of montant and two cross bars respectively.
As shown in fig. 6, prismatic pair constraint sets up as follows：Prismatic pair constraint need to ensure that thin bar GH can the free skating in thick bar EF
It is dynamic, at the same thin bar and thick bar when truss deploys optional position in quadrangle ABCD.Its specific constraint equation sees below formula：
Wherein Φ3Constrained for prismatic pair.Φ3Ensure that E points are overlapped with A points, C points are overlapped with G points, wherein rA、rE、rCAnd rGPoint
Wei not position coordinateses of point A, E, C and the G under global coordinate system.Φ4Ensure that two braces can be slided mutually, whereinWithRespectively point E, H, F and G is along brace direction vector, and l is brace length.Φ4In last
It is this pattern Φ in order to ensure two braces in plane ABCD3With Φ4Just constitute the constraint equation of prismatic pair, so far, Φ1、
Φ2And Φ3Constitute the constraint equation vector of network model.
Step 105：According to the grid cell type of step 102-104 acquisitions, generalized coordinates vector, generalized velocity vector sum
Constraint equation, Equations of Motion of Multibody Systems is set up based on lagrange equations of the first kind, as follows：
Wherein M is mass of system battle array,It is vectorial for generalized acceleration,It is second dervative of the generalized coordinates vector to the time,
F () is the elastic force vector of system, and Q () is vectorial for the generalized external force of system, Φ () and Φq() is respectively step
The constraint equation vector sum constraint equation vector of 104 network models obtained is to the partial derivative matrix of generalized coordinates vector, ()q
For the partial derivative of function pair generalized coordinates vector, λ is Lagrange multiplier vector, and t is simulation time, t ∈ [0, T], and T is matrix
Transposition.
Step 106：Using broad sense-alpha implicit time integration algorithms, the Equations of Motion of Multibody Systems built is carried out
Time domain discrete, obtains the Equations of Motion of Multibody Systems after time domain discrete, using Newton-Rapson iteration to many after discrete
System system kinetics equation is iterated, and carries out Parallel implementation, output descriptor to the system of linear equations generated in iterative process
Coordinate vector q and Lagrange multiplier vector λ.
As shown in fig. 7, step 601-603 describes the step 106, it is specially：
Step 601：Equations of Motion of Multibody Systems is obtained according to step 105, using broad sense-alpha implicit time integrations
Algorithm carries out following discrete：
Wherein
In formula (7), β and γ take γ >=1/2 and β >=(1/2+ γ) to determine the algorithm parameter of computational accuracy and efficiency2/
4.N is iterations, and h is time integral step-length, for dissipative system high frequency response, introduces new algorithm vector parameters a, wherein
Vector parameters a meets relation：
The choosing method of each parameter is as follows in formula (7) and formula (8):
WhereinFor the spectral radius of algorithm, the frequency range of algorithm energy dissipation distribution is decide, is generally takenαmAnd αfIt is broad sense-alpha implicit time integration algorithm parameters.
Step 602：Bring formula (7) into formula (6) and obtain the Nonlinear System of Equations of discrete form, and Nonlinear System of Equations is entered
Row solution obtains generalized coordinates vector qn+1, generalized velocity vectorWith Lagrange multiplier vector λ；
Step 603：Time t is judged according to step 602n+1Whether total simulation time T is less than, if current time tn+1It is not less than
Total simulation time T, then calculate and stop；Otherwise, n adds 1 automatically, makes time tn+1=tn+ h repeats step 601-603, calculates
Generalized coordinates vector qn+1, generalized velocity vectorUntil tn+1More than or equal to T.Output descriptor coordinate vector qn+1And draw
Ge Lang multipliers vector λn+1, wherein qn+1For q, λn+1For λ.
As shown in figure 8, step 6021-6025 has been further described through step 602, it is specially：
Step 6021：Solving precision Tol is set；
Step 6022：Using Newton-Rapson iteratives formula (6), following linear algebraic equation systems are obtained：
Wherein Δ q and Δ λ are the increments of current time t system generalized coordinates vector sum Lagrange multiplier vector, wherein
WithTo meet the algorithm parameter of following relation:
Wherein,WithIt is intermediate variable, ψ and Γ are intermediary matrix,For the inclined of function pair generalized velocity vector
Derivative；
Step 6023：The system of linear equations obtained according to step 6021 calculates current residue vector field homoemorphism r；
Wherein, | | | | it is mould；
Step 6024：Current generalized coordinates vector is updated to the solution of system of linear equations according to step 6022 and glug is bright
Day multiplier vector；
Step 6025：Judge whether iteration result restrains according to the residual vector mould that step 6023 is obtained, if r>Tol, then
The generalized coordinates vector and Lagrange multiplier vector obtained is updated according to step 6024, step 6022-6025 is repeated,
Until r<Tol or r=Tol, generalized coordinates vector q and Lagrange multiplier vector λ that output step 6024 is obtained, and deposit
Storage.So far complete to certain annular truss rope net antenna mechanism expansion Dynamic Modeling and efficiently calculating.Fig. 9 gives certain annular purlin
The system configuration at 6 specified moment during the expansion of frame antenna, the figure is substantially illustrated during the expansion of annular truss antenna
The corresponding ground test result of asynchrony phenomenon demonstrates the validity of the Numerical Simulation Results.
In summary, presently preferred embodiments of the present invention is these are only, is not intended to limit the scope of the present invention.
Within the spirit and principles of the invention, any modification, equivalent substitution and improvements made etc., should be included in the present invention's
Within protection domain.
Claims (5)
1. a kind of spatial complex flexible structure dynamics of multibody systems modeling and computational methods, it is characterised in that this method includes：
Step 101：Build threedimensional model and assign parameter：According to the material object of required solution, the threedimensional model in kind is set up, is inputted
Material parameter, geometric parameter and the space structure topological relation of institute's established model；
Step 102：Mesh generation：The threedimensional model after parameter is assigned based on step 101 and carries out mesh generation, absolute node is utilized
Coordinate method, geometry Precise equation wait geometric analysis method to determine grid cell type, and the threedimensional model after division is defined as into net
Lattice model；
Step 103：The grid divided according to step 102 determines start node position and the start node speed of each grid cell
Degree, grid cell generalized coordinates vector q and generalized velocity vector are calculated based on unit start node position and start node speed
Step 104：The generalized coordinates vector sum generalized velocity vector obtained according to step 103, sets the outside load of grid model
Lotus and boundary condition, the boundary condition based on grid model set up the constraint equation of grid model, and set emulation total duration T
And simulation time step-length h；
Step 105：According to the grid cell type of step 102-104 acquisitions, generalized coordinates vector, the constraint of generalized velocity vector sum
Equation, Equations of Motion of Multibody Systems is set up based on lagrange equations of the first kind, as follows：
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Wherein M is mass of system battle array,For generalized acceleration vector, F () is the elastic force vector of system, and Q () is system
Generalized external force vector, Φ () and Φq() is respectively the constraint equation vector sum constraint equation vector of step 104 acquisition to wide
The partial derivative matrix of adopted coordinate vector, ()qFor the partial derivative of function pair generalized coordinates vector, λ is Lagrange multiplier vector,
T is simulation time, and t ∈ [0, T], T is matrix transposition；
Step 106：Time domain discrete is carried out to the Equations of Motion of Multibody Systems built, the multi-body system after time domain discrete is obtained and moves
Mechanical equation, is iterated to the Equations of Motion of Multibody Systems after discrete, and the system of linear equations to being generated in iterative process
Parallel implementation, output descriptor coordinate vector q and Lagrange multiplier vector λ are carried out, so far completes to tie spatial complex flexibility
The modeling and solution of structure dynamics of multibody systems problem.
2. a kind of spatial complex flexible structure dynamics of multibody systems is modeled and efficient computational methods as claimed in claim 1, its
It is characterised by, the step 106 is specially：
Step 601：Equations of Motion of Multibody Systems is obtained according to step 105, to Equations of Motion of Multibody Systems carry out time domain from
Dissipate, discrete rear Equations of Motion of Multibody Systems is shown in formula (2)；
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Step 602：Equations of Motion of Multibody Systems after discrete is carried out solving the generalized coordinates vector q for obtaining the (n+1)th stepn+1、
Generalized velocity vectorWith Lagrange multiplier vector λn+1；
Step 603：Time t is judged according to step 602n+1Whether total simulation time T is less than, if current time tn+1It is imitative not less than total
True time T, then calculate and stop；Otherwise, n adds 1 automatically, makes time tn+1=tn+ h, repeats step 601-603, calculates broad sense
Coordinate vector qn+1, generalized velocity vectorUntil tn+1More than or equal to T, output descriptor coordinate vector qn+1And glug is bright
Day multiplier vector λn+1, wherein qn+1For q, λn+1For λ.
3. a kind of spatial complex flexible structure dynamics of multibody systems is modeled and efficient computational methods as claimed in claim 1 or 2,
Characterized in that, carrying out discrete adopt with the following method to Equations of Motion of Multibody Systems：
Equations of Motion of Multibody Systems is carried out using broad sense-alpha implicit time integrations algorithm following discrete：
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<mtd>
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<mo>-</mo>
<mo>-</mo>
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<mo>(</mo>
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Wherein
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<mo>(</mo>
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<mn>1</mn>
<mo>-</mo>
<mi>&gamma;</mi>
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<msub>
<mi>a</mi>
<mi>n</mi>
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<mo>+</mo>
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<mi>h&gamma;a</mi>
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</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula (3), β and γ is determine the algorithm parameter of computational accuracy and efficiency, and n is iterations, and h is simulation time step-length,
For dissipative system high frequency response, introduce new algorithm vector parameters a, wherein vector parameters a and meet relation：
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<mtd>
<mrow>
<mo>(</mo>
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<mi>m</mi>
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<mo>=</mo>
<mo>(</mo>
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<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
The choosing method of each parameter is as follows in formula (3) and formula (4):
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<mtd>
<mrow>
<msub>
<mi>&alpha;</mi>
<mi>m</mi>
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<mfrac>
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<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
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<mi>&alpha;</mi>
<mi>f</mi>
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</mrow>
<mn>2</mn>
</msup>
<mo>,</mo>
<mi>&gamma;</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
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<mi>&alpha;</mi>
<mi>f</mi>
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<mi>&alpha;</mi>
<mi>m</mi>
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</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
WhereinFor the spectral radius of algorithm, αmAnd αfIt is broad sense-alpha implicit time integration algorithm parameters.
4. a kind of spatial complex flexible structure dynamics of multibody systems is modeled and efficient computational methods as claimed in claim 2, its
It is characterised by, step 602 detailed process is：
Q is represented using qn+1, λ is represented using λn+1；
Step 6021：Solving precision Tol is set；
Step 6022：Using formula (2) described in Newton-Rapson iteratives, following linear algebraic equation systems are obtained：
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>&psi;</mi>
</mtd>
<mtd>
<msubsup>
<mi>&Phi;</mi>
<mi>q</mi>
<mi>T</mi>
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</mtd>
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<mtr>
<mtd>
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<mi>q</mi>
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</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
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</mfenced>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>&Delta;</mi>
<mi>q</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>&Delta;</mi>
<mi>&lambda;</mi>
</mtd>
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<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
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<mi>&Gamma;</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>&Phi;</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein Δ q and Δ λ are the increments of current time t system generalized coordinates vector sum Lagrange multiplier vector, wherein
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<mtable>
<mtr>
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<mrow>
<mi>&psi;</mi>
<mo>=</mo>
<mi>M</mi>
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<mi>q</mi>
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</mover>
<mo>+</mo>
<msubsup>
<mi>&Phi;</mi>
<mi>q</mi>
<mi>T</mi>
</msubsup>
<mi>&lambda;</mi>
<mo>+</mo>
<mi>F</mi>
<mrow>
<mo>(</mo>
<mi>q</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>q</mi>
<mo>,</mo>
<mover>
<mi>q</mi>
<mo>&CenterDot;</mo>
</mover>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
WithTo meet the algorithm parameter of following relation:
<mrow>
<mover>
<mi>&beta;</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mi>m</mi>
</msub>
</mrow>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
<mi>&beta;</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mi>f</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mover>
<mi>&gamma;</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<mfrac>
<mi>&gamma;</mi>
<mrow>
<mi>h</mi>
<mi>&beta;</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,WithIt is intermediate variable, ψ and Γ are intermediary matrix,For the local derviation of function pair generalized velocity vector
Number；
Step 6023：The system of linear equations obtained according to step 6021 calculates current residue vector field homoemorphism r；
<mrow>
<mi>r</mi>
<mo>=</mo>
<mo>|</mo>
<mo>|</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>&Gamma;</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>&Phi;</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>|</mo>
<mo>|</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, | | | | it is mould；
Step 6024：Current generalized coordinates vector is updated to the solution of system of linear equations according to step 6022 and Lagrange multiplies
Subvector；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>q</mi>
<mo>=</mo>
<mi>q</mi>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>q</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>&lambda;</mi>
<mo>=</mo>
<mi>&lambda;</mi>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>&lambda;</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Step 6025：Judge whether iteration result restrains according to the residual vector mould that step 6023 is obtained, if r>Tol, then basis
Step 6024 updates the generalized coordinates vector and Lagrange multiplier vector obtained, repeats step 6022-6025, until
r<Tol or r=Tol, generalized coordinates vector q and Lagrange multiplier vector λ that output step 6024 is obtained.
5. a kind of spatial complex flexible structure dynamics of multibody systems is modeled and efficient computational methods as claimed in claim 3, its
It is characterised by, it is described
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