CN106407588A - Simulated analysis platform for thermal disturbance responses of spacecraft - Google Patents

Abstract

The invention belongs to the technical field of design, and dynamics simulation and control of high-precision spacecraft, and in particular relates to a simulated analysis platform for thermal disturbance responses of the spacecraft. The platform comprises a data input modeling module, a flexible attachment on-orbit thermal analysis module, an equivalent thermal load derivation module, a modal analysis module, a coupling dynamics modeling module, a coupling dynamics model solution module and a post-processing module in sequence; the simulated analysis platform builds a coupling dynamics model of large flexible satellite-borne attachments and stars, and provides a special simulated analysis platform for coupling responses of the large flexible satellite-borne attachments and the stars. The simulated analysis platform can be used for solving posture variations of the spacecraft with complex flexible attachments caused by thermally induced dynamic deformations of the flexible attachments, can provide a simple and effective model, method and tool for forecast and assessment of thermally induced vibration responses of the spacecraft with the flexible attachments under actual conditions, and can serve as a reference for a satellite design process.

Description

Spacecraft thermal agitation responds Simulation Platform

Technical field

The invention belongs to high accuracy Spacecraft guidance and control and dynamics simulation and control technology field, specially spacecraft heat is disturbed Dynamic response Simulation Platform.

Background technology

When spacecraft turnover earth's shadow area, spatial heat environment changes, and the acute variation of temperature not only can make soft Property adnexa there is larger thermal deformation, induce thermal vibration, and perturbed force effect also can be delivered in spacecraft main body.Due to angle The conservation of momentum, the vibration of flexible accessory can lead to spacecraft subjective posture to be shaken, and then affects spacecraft payload Pointing accuracy and attitude stability, lead to the spacecraft cannot normal work or disabler.

NASA has had observed that the impact to spacecraft orbit attitude for the heat-driven oscillation, also delivered in the world many because Flexible accessory occurs heat-driven oscillation to cause the example of spacecraft operational failure.With later batch scholar to Thermal Load to space flight The impact of device attitude dynamics has been made to study.Corresponding Correlation Analysis Technique has carried out substantial amounts of research work from component-level angle Make.But most of document consider be all adnexa the impact to the attitude of satellite for the quasistatic thermal deformation, that is, have ignored thermal response Transient term, can not embody in a model because of the track hot induced vibration problem that day alternates with night leads to.

Analyze the impact to attitude motion of spacecraft for the flexible structure thermally-induced motion from total system aspect, both domestic and external Research work is not very abundant.Flexible accessory thermic dynamic deformation induces analysis and the emulation of spacecraft attitude change, is related to Many calculating such as the in-orbit Orbital heat flux/ascent of flexible accessory, transient state temperature field, mode, dynamic deformation, coupling response, belong to In typical multi-crossed disciplines problem, single simulation analysis system completes to calculate extremely difficult, the special system of simulation software of analysis System also lacks very much.Relatively new development in spacecraft rigid body-adnexa coupled system heat-dynamics research field is as number Johnston Research with Thornton.Johnston and Thornton heat-non-coupled method of structure discusses flexible accessory heat-driven oscillation Impact to spacecraft dynamic response, the satellite system simplified model being made up of Rigid Base and flexible accessory for a class, send out Open up a kind of heat-Structural Dynamicses uncoupled two dimensional surface motion theory analysis method, have studied the heat of satellite flexible accessory The impact to satellite plane attitude dynamics for the induced movement.Weak point is to regard simple beam as to process by flexible accessory, for The labyrinth form of actual adnexa discusses.

Higher spacecraft is required for pointing accuracy and attitude stability, if only with simplified model come approximately complexity Flexible accessory structure is it is clear that be not suitable for flexible accessory thermic dynamic deformation is induced with the Interaction Mechanics characteristic of spacecraft attitude change Carry out simulation analysis it is impossible to Accurate Prediction and assessment the impact to spacecraft attitude for the spaceborne flexible member heat-driven oscillation.Therefore, compel It is essential and will set up the theory that can solve the response of the spacecraft thermal agitation with complicated flexible accessory structure and numerical analysis model, development Corresponding solution technique and software system.

Content of the invention

For above-mentioned technical problem, the purpose of the present invention is for solving the induction space flight of large-scale flexible adnexa thermic dynamic deformation The analysis of device attitudes vibration and emulation, the spacecraft thermal agitation response that a kind of spaceborne large-scale flexible adnexa of proposition is coupled with celestial body is specially Use Simulation Platform.

Design principle is：For Rigid Base-flexible accessory class spacecraft, using this kind of underexcitation of equivalent heat load as disturbing Dynamic source, carries out modelling, the full star kinetic energy of foundation meter and hot load action and gesture using hybrid coordinate method and FInite Element to system Energy model, sets up spacecraft thermic micro-vibration Coupling Dynamic Model using Lagrange method.After numerical discretization, coupling Kinetic model is eventually exhibited as one group of nonlinear equation, recycles Newmark method to be solved with reference to Newton iteration method.This Inventing targeted object of study is the spacecraft with flexible accessory.Compared to flexible accessory, the rigidity of spacecraft center nacelle Much bigger, therefore center nacelle can be approximately the rigid body with lumped mass and rotary inertia, thus whole spacecraft As Rigid Base-flexible accessory coupled system, and ignore the translation displacements of Rigid Base.

Specifically technical scheme is：

Spacecraft thermal agitation responds Simulation Platform, includes data input MBM, the in-orbit heat of flexible accessory successively Analysis module, equivalent heat load lead calculation module, model analyses module, Coupled Dynamics MBM, Coupling Dynamic Model are asked Solution module and post-processing module；

(1) data input MBM：Set up spacecraft Rigid Base-soft using interactive mode with reference to automatic conversion mode The FEM (finite element) model of property adnexa coupled system and in-orbit thermal model；

(2) the in-orbit thermal analysis module of flexible accessory：Existed using the spacecraft Rigid Base-flexible accessory coupled system set up Rail thermal model, carries out the in-orbit heat analysis of Flexible appendages of spacecraft, obtains the transient state temperature field on flexible accessory；

The heat transfer type that the in-orbit heat analysis of flexible accessory structure are related to is mainly conduction of heat and heat radiation.The present invention series of fortified passes Note is that spacecraft passes in and out the flexible accessory vibration problem causing during earth's shadow due to temperature acute variation in the short time, because This, the Orbital heat flux suffered by flexible accessory mainly considers solar radiation hot-fluid.In-orbit thermoanalytical conduction of heat fundamental equation with logical Under normal radiation heat transfer, the equation of heat conduction is identical, but increased orbit computation, ascent calculating, the calculating of Orbital heat flux.

(3) equivalent heat load leads calculation module：Carry out the equivalent heat of transient state temperature field on flexible accessory using initial strain method Load leads calculation, obtains time dependent equivalent nodal force and moment on each node；

Using the FEM (finite element) model of the spacecraft Rigid Base-flexible accessory coupled system set up, by wink on flexible accessory The hot load in state temperature field is equivalent to the joint load on flexible accessory.Boom, beam, plate shell is generally comprised in flexible accessory structure Deng component, need for suffered temperature change on these member units to be equivalent to time dependent nodal force load, so that Carry out the coupled system micro-vibration time-histories data analysis of next step.Using initial strain (temperature strain) method in Finite Element Method To calculate the equivalent nodal force of temperature load.According to the deformation pattern of unit, the temperature load of boom unit can be equivalent to axle Xiang Li, the temperature load of beam element can be equivalent to axial force and moment, and the temperature load of Shell Finite Element can be equivalent in face Power dough-making powder moment of face, can complete Equivalent Calculation by the integration in unit.

(4) model analyses module：Using structural finite element model, using iteration WYD-Ritz vector, directly Superposition Method is carried out The model analyses of Rigid Base-flexible accessory coupled system, obtain cycle and the vibration shape of coupled system；

Carry out the mould of spacecraft Rigid Base-flexible accessory coupled system using the direct Superposition Method of iteration WYD-Ritz vector State is analyzed.Using packet shift frequency, mode error convergence criterion, the multinomial technology such as sparse Fast Direct Method of cell improve efficiency, Solving precision and reliability.The scale of solving a problem of current eigenvalue problem on common computer ten thousand degree of freedom up to 30 to 50, can Accurately solve up to hundreds of low side mode.Mode error convergence criterion makes the process of model analyses become steady.Test knot Fruit shows, mode error bit value indicative error more can reflect the precision that eigenvalue problem calculates.When calculating more multi-modal, mode Error should be used as first-selected convergence criterion.

(5) Coupled Dynamics MBM：Using modal expanding and Lagrange equation, according to finite element data and equivalent Hot payload data, sets up the thermic micro-vibration Coupling Dynamic Model of spacecraft coupled system；

Flexible accessory with the kinetic energy expression of celestial body coupled system is：

In formula, M is Space Vehicle System quality；With modal coordinate, mode exhibition is carried out to the malformation of flexible accessory in above formula After opening, Flexible appendages of spacecraft with the kinetic energy equation of celestial body coupled system is：

In formula, Φ, η are respectively modal matrix and the modal coordinate battle array of flexible accessory structure；

For Space Vehicle System quiet away from；

For Space Vehicle System relative to barycenter rotary inertia Battle array；

For the flexibility to spacecraft translation for the flexible accessory structure heat-driven oscillation Coefficient of coup matrix；

For flexible accessory structure heat-driven oscillation, spacecraft is rotated Flexible couplings coefficient matrix；

Rigid body mode battle array for flexible accessory structure；

After group collection flexible accessory structure all units strain energy, then the spacecraft potential energy equation of meter and hot load effect is：

K, r in formula_TIt is respectively the Stiffness Matrix of flexible accessory structure and hot load battle array, Λ is that adnexa is firm in Modal Space Degree battle array；

The Lagrange function representation of Space Vehicle System is：

According to the Lagrange equation of nonholonomic nonconservative, do not consider the translation of spacecraft, and ignore some higher order terms, then carry The coupled dynamical equation of the spacecraft Rigid Base-flexible accessory coupled system of single flexible adnexa is finally reduced to：

Coefficient matrix in above formula can utilize the geological information of coupled system FEM (finite element) model, mass matrix and intrinsic Frequency and the vibration shape etc. obtain, and hot load equivalent nodal force can also utilize initial strain method by equivalent after obtaining temperature field data Hot load is led calculation module and is obtained.

(6) Coupling Dynamic Model solves module：Combine Newton iteration method using Newmark method, carry out spacecraft coupling The solution of the thermic micro-vibration Coupling Dynamic Model of assembly system, obtains time-histories data result and the spacecraft appearance of flexible accessory The time-histories data result at state angle；

The coupled dynamical equation is rewritten as：

The state variable of definition system is：

Then the coupled dynamical equation is represented by：

Discrete in time domain using Newmark method, its numeric format is：

Wherein,

As δ >=0.5, γ >=0.25 (0.5+ δ)²When, algorithm is unconditional stability, so allow for using larger when Between step-length, for example elect the one of some points of the structure minimum period as.Therefore Newmark method can be used to solve time-histories longer Time-histories data, and larger time step can also omit the impact that the inaccurate characteristic solution of high-order responds to system.In the present invention In, select δ=0.5, γ=0.25.

The coupled dynamical equation is Nonlinear System of Equations, needs to be changed using Newton iteration method within each time step In generation, solves, and its Iteration is：

Above formula is substituted in load item, and it is decomposed into two parts：

Wherein,Require unknown quantity q with current time step_n+1It is relevant,Relevant with a upper time step known quantity.Again After arrangement, it is denoted as：

Ψ(q_n+1)=F

Using Newton iteration method, the iterative formula that it solves is：

In Practical Calculation, only need to iteration in each time step and can restrain for 3-5 time.

When solving Dynamics Coupling equation, due to employing mode superposition method, after obtaining the modal information of model Greatly reduce the exponent number of equation group, adopt the Newmark method of unconditional stability to solve dynamic response it is allowed to use relatively simultaneously Big time step, contributes to simulating the longer time or solves more massive problem, restrain in the iteration of each step Also quickly, the efficient of derivation algorithm and reliability are illustrated.

(7) post-processing module：The temperature variation curve extract and show, exporting each node of flexible accessory, equivalent load become Change the result of calculations such as curve, micro-vibration time-histories data curve and spacecraft attitude angle change curve.

Post-processing module mainly requires to extract and show, export the result of calculation of correlation according to user, for example flexible attached The temperature variation curve of each node on part, equivalent load change curve, flexible accessory micro-vibration time-histories data curve, spacecraft Attitude angle change curve.

The spacecraft thermal agitation response Simulation Platform that the present invention provides, is carried on the back with high pointing accuracy spacecraft for engineering Scape, establishes spaceborne large-scale flexible adnexa and celestial body Coupling Dynamic Model it is proposed that a kind of spaceborne large-scale flexible adnexa and star The dedicated emulated analysis platform of body coupling response.This Simulation Platform can be used to solve band complexity flexible accessory spacecraft by The attitudes vibration that its flexible accessory thermic dynamic deformation is induced, can shake for practical situation lower band flexible accessory spacecraft thermic The prediction of dynamic response and assessment provide a kind of simple and effective model, Method and kit for, can be used as real satellite design process In reference.

Brief description

Fig. 1 is present configuration schematic diagram；

Fig. 2 is that the LEO spacecraft of embodiment simplifies structural representation；

Fig. 3 is end angle point side to light, shady face temperature variation curve and its difference variation curve of embodiment；

Fig. 4 is the equivalent moment curve of the endpoint node of embodiment；

Fig. 5 is the end angle point displacement of the lines vertically of embodiment；

Fig. 6 is the end angle point of embodiment around axial angular displacement；

Fig. 7 is the Rigid Base attitude angle change curve of embodiment.

Specific embodiment

In order to further illustrate objects and advantages of the present invention, with instantiation, the present invention is made into one below in conjunction with the accompanying drawings The explanation of step.

The spacecraft thermal agitation response Simulation Platform structure of the present embodiment is as shown in figure 1, include data input successively The in-orbit thermal analysis module of MBM, flexible accessory, equivalent heat load lead calculation module, model analyses module, Coupled Dynamics are built Mould module, Coupling Dynamic Model solve module and post-processing module.

It is object of study using Low Earth Orbit (LEO) spacecraft with single-blade sun battle array structure as shown in Figure 2.Pin To the flexible sun battle array structure dynamics deformation causing from sunshine area entrance shadow region acute variation thermal environment and the vibration of celestial body attitude Response has carried out Numerical Simulation Analysis.

Using platform proposed by the present invention, implement numerical simulation according to the following steps：

(1) FEM (finite element) model of coupled system and in-orbit thermal model are set up using data input MBM；

(2) carry out the in-orbit heat analysis of solar battery array using the in-orbit thermal analysis module of flexible accessory, obtain solar cell Battle array flexible structure in-orbit period is especially into and out transient temperature field data during earth's shadow area；

(3) lead calculation module using equivalent heat load, the transient temperature field data that previous step is obtained is equivalent to be applied to Hot load on solar battery array flexible structure node；

(4) utilize model analyses module, coupling system model is carried out with model analyses, obtain front 20 order mode states；Whole boat Its device, as Rigid Base-flexible accessory coupled system, only constrains three translational degree of freedom of central point；

(5) utilize Coupled Dynamics MBM, the finite element data according to coupled system and equivalent heat payload data, build The thermic micro-vibration Coupling Dynamic Model of vertical spacecraft coupled system；

(6) utilize Coupling Dynamic Model to solve module, carry out the thermic micro-vibration Coupling Dynamic Model of coupled system Solution, obtain the time-histories data result of flexible accessory and the time-histories data result at spacecraft attitude angle.

Fig. 3 is that solar battery array end angle point is entering earth's shadow moment corresponding temperature variation curve and difference variation Curve.From Fig. 3 it is found that solar battery array structure difference variation curve after initially entering earth's shadow through about The time of 20s trends towards stable.Therefore, can assert that the heat structure response time of this solar battery array is approximately 20s, simultaneously its knot Structure response time (first step mode corresponding cycle) is 8.63s, and the two is more or less the same.This flexible sun battle array structure is entering ground It is likely to occur thermic dynamic deformation during ball shade.

Fig. 4 is the equivalent moment curve of this solar battery array end angle point.To illustrate the 0s moment as original state, and suppose Displacement now, equivalent heat load are zero.Sun battle array end angle point can be obtained by the coupled dynamical equation solution and center is firm Body attitude corresponding thermic response curve, is shown in Fig. 5, Fig. 6 and Fig. 7 respectively.

Knowable to above-mentioned simulation analysis result：The vibration that the acute variation of spatial heat environment causes this satellite system rings Should.Wherein, sun battle array structure angle point axial deformation maximum amplitude be 0.95mm, flexural deformation maximum amplitude in 0.54arcsec, Celestial body attitude angle produces the maximum change close to 0.042arcsec, and solar battery array thermic micro-vibration response level is less, but Induce the tremor response of the attitude of satellite.

Claims (3)

1. spacecraft thermal agitation response Simulation Platform is it is characterised in that include data input MBM, flexible attached successively The in-orbit thermal analysis module of part, equivalent heat load lead calculation module, model analyses module, Coupled Dynamics MBM, coupling power Learn model solution module and post-processing module；

(1) data input MBM：Set up spacecraft Rigid Base-flexibility using interactive mode with reference to automatic conversion mode attached The FEM (finite element) model of part coupled system and in-orbit thermal model；

(2) the in-orbit thermal analysis module of flexible accessory：Using the spacecraft Rigid Base-in-orbit heat of flexible accessory coupled system set up Analysis model, carries out the in-orbit heat analysis of Flexible appendages of spacecraft, obtains the transient state temperature field on flexible accessory；

(3) equivalent heat load leads calculation module：Carry out the equivalent heat load of transient state temperature field on flexible accessory using initial strain method Lead calculation, obtain time dependent equivalent nodal force and moment on each node；

(4) model analyses module：Using structural finite element model, using iteration WYD-Ritz vector, directly Superposition Method carries out center The model analyses of rigid body-flexible accessory coupled system, obtain cycle and the vibration shape of coupled system；

(5) Coupled Dynamics MBM：Using modal expanding and Lagrange equation, according to finite element data and equivalent heat lotus Carry data, set up the thermic micro-vibration Coupling Dynamic Model of spacecraft coupled system；

(6) Coupling Dynamic Model solves module：Combine Newton iteration method using Newmark method, carry out spacecraft coupled systemes The solution of the thermic micro-vibration Coupling Dynamic Model of system, obtains time-histories data result and the spacecraft attitude angle of flexible accessory Time-histories data result；

(7) post-processing module：The temperature variation curve extract and show, exporting each node of flexible accessory, equivalent load change are bent The result of calculations such as line, micro-vibration time-histories data curve and spacecraft attitude angle change curve.

2. spacecraft thermal agitation according to claim 1 responds Simulation Platform it is characterised in that described coupling is moved The thermic micro-vibration Coupling Dynamic Model that mechanical modeling module sets up spacecraft coupled system is specially：

Flexible accessory with the kinetic energy expression of celestial body coupled system is：

$\begin{array}{c}T=\frac{1}{2}{\stackrel{\·}{X}}^{T}M\stackrel{\·}{X}+{\stackrel{\·}{X}}^T(\underset{A}{\Σ}{\mathrm{dm}}_a\[{\tilde{d}}_b^T+C_b^{aT}({\tilde{r}}_a^T+{\widetilde{\mathrm{\δ}}}_a)C_b^a\]+\underset{B}{\Σ}{\mathrm{dm}}_b{{\tilde{r}}_b}^T){\mathrm{\ω}}_b+\\ {\stackrel{\·}{X}}^T\underset{A}{\Σ}{\mathrm{dm}}_a{C}_b^{aT}{\stackrel{\·}{\mathrm{\δ}}}_{aT}+\frac{1}{2}{\mathrm{\ω}}_b^T\left(\underset{A}{\Σ}{\mathrm{dm}}_a{\left(C_b^a{\tilde{d}}_b^T+{\tilde{r}}_a^T{C}_b^a\right)}^T\right(C_b^a{\tilde{d}}_b^T+{\tilde{r}}_a^T{C}_b^a)+\underset{B}{\Σ}{\mathrm{dm}}_b{\tilde{r}}_b{{\tilde{r}}_b}^T){\mathrm{\ω}}_b+\\ {\mathrm{\ω}}_b^T\underset{A}{\Σ}{\mathrm{dm}}_a{\left(C_b^a{\tilde{d}}_b^T+{\tilde{r}}_a^T{C}_b^a\right)}^T{\stackrel{\·}{\mathrm{\δ}}}_{aT}+\frac{1}{2}\underset{A}{\Σ}{\mathrm{dm}}_a{\stackrel{\·}{\mathrm{\δ}}}_a^T{\stackrel{\·}{\mathrm{\δ}}}_{aT}\end{array}$

In formula, M is Space Vehicle System quality；With modal coordinate, the malformation of flexible accessory in above formula is carried out after modal expanding, Flexible appendages of spacecraft with the kinetic energy equation of celestial body coupled system is：

$T=\frac{1}{2}{\stackrel{\·}{X}}^{T}M\stackrel{\·}{X}+{\stackrel{\·}{X}}^T{\mathrm{P\ω}}_b+{\stackrel{\·}{X}}^{T}R\stackrel{\·}{\mathrm{\η}}+\frac{1}{2}{\mathrm{\ω}}_b^T{\mathrm{I\ω}}_b+{\mathrm{\ω}}_b^{T}F\stackrel{\·}{\mathrm{\η}}+\frac{1}{2}{\stackrel{\·}{\mathrm{\η}}}^T\stackrel{\·}{\mathrm{\η}}$

In formula, Φ, η are respectively modal matrix and the modal coordinate battle array of flexible accessory structure；

For Space Vehicle System quiet away from；

For Space Vehicle System relative to barycenter rotary inertia battle array；

For the flexible couplings system to spacecraft translation for the flexible accessory structure heat-driven oscillation Matrix number；

For flexible accessory structure heat-driven oscillation spacecraft is rotated soft Property coefficient of coup matrix；

Rigid body mode battle array for flexible accessory structure；

After group collection flexible accessory structure all units strain energy, then the spacecraft potential energy equation of meter and hot load effect is：

$U=\ΣU_e=\frac{1}{2}{\mathrm{\δ}}^{T}K\mathrm{\δ}-{\mathrm{\δ}}^T{r}_T=\frac{1}{2}{\mathrm{\η}}^T\mathrm{\Λ}\mathrm{\η}-{\mathrm{\η}}^T\left({\mathrm{\Φ}}^T{r}_T\right)$

K, r in formula_TIt is respectively the Stiffness Matrix of flexible accessory structure and hot load battle array, Λ is Stiffness Matrix in Modal Space for the adnexa；

The Lagrange function representation of Space Vehicle System is：

$\begin{array}{c}L=T-V\\ =\frac{1}{2}{\stackrel{\·}{X}}^{T}M\stackrel{\·}{X}+{\stackrel{\·}{X}}^{T}R\stackrel{\·}{\mathrm{\η}}+\frac{1}{2}{\mathrm{\ω}}_b^T{\mathrm{I\ω}}_b+{\mathrm{\ω}}_b^{T}F\stackrel{\·}{\mathrm{\η}}+\frac{1}{2}{\stackrel{\·}{\mathrm{\η}}}^T\stackrel{\·}{\mathrm{\η}}-\frac{1}{2}{\mathrm{\η}}^T\mathrm{\Λ}\mathrm{\η}+{\mathrm{\η}}^T\left({\mathrm{\Φ}}^T{r}_T\right)\end{array}$

According to the Lagrange equation of nonholonomic nonconservative, do not consider the translation of spacecraft, and ignore some higher order terms, then carry single The coupled dynamical equation of the spacecraft Rigid Base-flexible accessory coupled system of flexible accessory is finally reduced to：

$\left\{\begin{array}{c}I{\stackrel{\·}{\mathrm{\ω}}}_b+{\widetilde{\mathrm{\ω}}}_{b}I{\mathrm{\ω}}_b+F\stackrel{\·\·}{\mathrm{\η}}+{\widetilde{\mathrm{\ω}}}_{b}F\stackrel{\·}{\mathrm{\η}}=0\\ \stackrel{\·\·}{\mathrm{\η}}+2\mathrm{\ξ}\mathrm{\Ω}\stackrel{\·}{\mathrm{\η}}+\mathrm{\Λ}\mathrm{\η}+F^T{\stackrel{\·}{\mathrm{\ω}}}_b={\mathrm{\Φ}}^T{r}_T\end{array}\right.$

Coefficient matrix in above formula can utilize geological information, mass matrix and the natural frequency of coupled system FEM (finite element) model Obtain with the vibration shape etc., hot load equivalent nodal force can also after obtaining temperature field data using initial strain method by equivalent heat lotus Load is led calculation module and is obtained.

3. spacecraft thermal agitation according to claim 1 responds Simulation Platform it is characterised in that described coupling is moved Mechanical model solves module, and solution procedure is：

The coupled dynamical equation is rewritten as：

$\left[\begin{array}{cc}I& F\\ F^T& E\end{array}\right]\left(\begin{array}{c}\stackrel{\·\·}{\mathrm{\θ}}\\ \stackrel{\·\·}{\mathrm{\η}}\end{array}\right)+\left[\begin{array}{cc}0& 0\\ 0& 2\mathrm{\ξ}\mathrm{\Ω}\end{array}\right]\left(\begin{array}{c}\stackrel{\·}{\mathrm{\θ}}\\ \stackrel{\·}{\mathrm{\η}}\end{array}\right)+\left[\begin{array}{cc}0& 0\\ 0& \mathrm{\Λ}\end{array}\right]\left(\begin{array}{c}\mathrm{\θ}\\ \mathrm{\η}\end{array}\right)=\left(\begin{array}{c}{P}_1\\ {\mathrm{\Φ}}^T{r}_T\end{array}\right)$

The state variable of definition system is：

$q=\left\{\begin{array}{c}\mathrm{\θ}\\ \mathrm{\η}\end{array}\right\}$

Then the coupled dynamical equation is represented by：

$M\stackrel{\·\·}{q}+C\stackrel{\·}{q}+Kq=P(q,\stackrel{\·}{q})$

Discrete in time domain using Newmark method, its numeric format is：

$\begin{array}{c}(K+c_{0}M+c_{1}C)q_{n+1}=M(c_0{q}_n+c_2{\stackrel{\·}{q}}_n+c_3{\stackrel{\·\·}{q}}_n)\\ +C(c_1{q}_n+c_4{\stackrel{\·}{q}}_n+c_5{\stackrel{\·\·}{q}}_n)+P_{n+1}(q_{n+1},{\stackrel{\·}{q}}_{n+1})\end{array}$

Wherein,

Select δ=0.5, γ=0.25；

The coupled dynamical equation is Nonlinear System of Equations, needs to be iterated asking using Newton iteration method within each time step Solve, its Iteration is：

${\stackrel{\·}{q}}_{n+1}=c_1{q}_{n+1}-c_1{q}_n-c_4{\stackrel{\·}{q}}_n-c_5{\stackrel{\·\·}{q}}_n$

Above formula is substituted in load item, and it is decomposed into two parts：

$\begin{array}{c}{P}_{n+1}(q_{n+1},{\stackrel{\·}{q}}_{n+1})=P_{n+1}(q_n,{\stackrel{\·}{q}}_n,{\stackrel{\·\·}{q}}_n,q_{n+1})\\ =P_{n+1}^a\left(q_{n+1}\right)+P_{n+1}^b(q_n,{\stackrel{\·}{q}}_n,{\stackrel{\·\·}{q}}_n)\end{array}$

Wherein,Require unknown quantity q with current time step_n+1It is relevant,Relevant with a upper time step known quantity；Rearrange Afterwards, it is denoted as：

Ψ(q_n+1)=F

Using Newton iteration method, the iterative formula that it solves is：

In Practical Calculation, only need to iteration in each time step and can restrain for 3-5 time.