CN106484984B - The spaceborne thermotropic micro-vibration of flexible accessory responds Simulation Platform - Google Patents
The spaceborne thermotropic micro-vibration of flexible accessory responds Simulation Platform Download PDFInfo
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Abstract
The present invention relates to the thermotropic micro-vibrations of spaceborne flexible accessory to respond Simulation Platform, belong to high-precision Spacecraft guidance and control and dynamics simulation and control technology field, it leads including data input modeling module, in-orbit thermal analysis module, equivalent heat load and calculates module, high-precision model analysis module, high-precision Response Analysis module, post-processing module, modules are sequentially connected, input of the output of previous module as next module；The analysis platform carries out the research and development of the analysis and simulation research and corresponding software for calculation of the spaceborne large-scale flexible attachment thermal vibration of practical application, has achieved the purpose that quick, the thermotropic micro-vibration of effective acquisition large-scale flexible component and its has responded with celestial body coupled vibrations；Meanwhile analyzing emulation platform can push the engineer application of spaceborne large-scale flexible component thermal deformation and thermotropic micro-vibration simulation analysis technology and examination to verify, and drive Related Supporting Technologies progress, for subsequent New Satellite Project R&D accumulation technology basis.
Description
Technical field
The present invention relates to the thermotropic micro-vibrations of spaceborne flexible accessory to respond Simulation Platform, belongs to high-precision Spacecraft guidance and control
And dynamics simulation and control technology field.
Background technique
With the rapid development of space technology, large-size pliable structure and flexible accessory are widely used in all kinds of spacecrafts,
Such as large-scale Deployable antenna, solar array, antenna mounting arm.These flexible structures and flexible accessory often have area big, long
And feature thin or long and thin, that quality is small, supporting form multiclass are similar to cantilever design, lateral stiffness is relatively small, frequency compared with
It is low.This kind of flexible structure is equivalent to by thermal shock when temperature quickly changes, and disabler even recurring structure is caused to destroy.
When spacecraft passes in and out earth's shadow area, the acute variation of temperature not only makes flexible accessory that biggish thermal deformation occur, and induces heat
Vibration, and perturbed force effect can be also transmitted in spacecraft main body, seriously affect the stabilization, posture, pointing accuracy of spacecraft
Deng.
Earth observation that China is developing and deep space exploration remote-sensing spacecraft, vast capacity telecommunication satellite etc. belong to height
Precision spacecraft.To ensure the reliable of the safety and observation that communicate, this kind of high-precision spacecraft is to pointing accuracy and stability
There is high requirement.With reference to external data, high-precision spacecraft vibration index to be achieved is: angular variation amplitude is less than
0.007arcsec (Hubble HST), acceleration are lower than 10-3G (international space station ISS).Therefore, spacecraft is carried out
The thermotropic micro-vibration response analysis simulation Technique Study of flexible accessory and corresponding Development of Software Platform are of great significance and are worth.
The task of the thermotropic micro-vibration response analysis simulation study of Flexible appendages of spacecraft is mainly analyzed by spatial heat environment
The thermotropic micro-vibration response of flexible accessory caused by changing.Its main research include: the modeling of flexible accessory in-orbit heat analysis with
Temperature calculating, the calculating of transient state temperature field equivalent heat load, the thermotropic micro-vibration modeling of spaceborne flexible accessory are with analysis and specially
With simulation analysis system exploitation etc..
In terms of the in-orbit heat analysis of space structure, have several research achievements and special-purpose software.It is equivalent in transient state temperature field
Hot load calculates aspect, although correlation theory comparative maturity, is also short of in software realization, lacks special height very much
Efficiency software module.By secondary development, existing business software has had the function that Steady-State Thermal Field is equivalent to hot load, but
For engineering structure fairly large and with more time step, since data volume is very big, existing software is to its transient state temperature
The Equivalent Calculation of degree field is then substantially all and can not be effectively treated, and is not able to satisfy calculating demand.It is formed after equivalent heat load, it can be with
Structure analysis software is called to carry out the spaceborne thermotropic micro-vibration response analysis of flexible accessory.Many business softwares can complete this point
Analysis.However, the limitation of calculation scale still remains.Such as the load of medium scale structure, thousands of steps is walked, only read
Access evidence and the time for exporting calculated result can not endure.Therefore, there is an urgent need to develop high efficiency, high-precision, engineerings
Practical dynamic response analysis software module to complete the thermotropic micro-vibration analysis of spaceborne flexible accessory, and expeditiously obtains
Take required result information.
The thermotropic the Dynamic Response emulation of Flexible appendages of spacecraft, due to being related to in-orbit Orbital heat flux/ascent, transient temperature
Various calculating such as field, mode, dynamic deformation belong to typical multi-crossed disciplines problem, complete using single simulation analysis system
At calculating, analysis is relatively difficult, generally requires exploitation integrated analysis system.Its advantage is that calculating each stage using each chief
As a result precision and efficiency is able to maximum consideration, the disadvantage is that needing to develop the routine interface between software, data exchange amount compares
Greatly, computational efficiency is relatively low.In addition it is also necessary to the specially transmitting of exploitation data and management, post-processing module and platform interface etc..
Summary of the invention
In view of the above technical problems, the purpose of the present invention is be the spaceborne large-scale flexible attachment of solution due to spatial heat environment
The simulation and analysis of thermotropic micro-vibration response caused by variation proposes a kind of thermotropic micro-vibration response of spaceborne large-scale flexible attachment
Analyze the integrated platform system of emulation.
The design principle of platform of the invention is: being analyzed using the non-coupled dynamic analysis method of heat-structure and model
The thermotropic micro-vibration of spaceborne flexible accessory responds: i.e. hypothesis temperature field, hot load, the influence between displacement field are unidirectional, first
The temperature field that additional hot-fluid causes variation is calculated, it is then equivalent to be added on flexible accessory at the hot load changed over time, then count
Calculate the time-histories data of flexible accessory.The targeted research object of the present invention is the spacecraft with flexible accessory.It is attached compared to flexible
The rigidity of part, spacecraft center cabin is much bigger, therefore can be approximately band lumped mass and rotary inertia center cabin
Rigid body, thus rigid body centered on entire spacecraft-flexible accessory coupled system.The coupled system is in space only constrains
3 translation displacements of the lines of heart rigid body.
Specific technical solution are as follows:
The spaceborne thermotropic micro-vibration of flexible accessory responds Simulation Platform, including data input modeling module, in-orbit heat point
Analysis module, equivalent heat load, which are led, calculates module, high-precision model analysis module, high-precision Response Analysis module, post-processing mould
Block, modules are sequentially connected, input of the output of previous module as next module；
(1) data input modeling module, and it is soft to establish spacecraft Rigid Base-in the way of interactive mode combination automatic conversion
The finite element model and in-orbit thermal model of property attachment coupled system.
(2) in-orbit thermal analysis module utilizes spacecraft Rigid Base-in-orbit heat analysis of flexible accessory coupled system of foundation
Model carries out the in-orbit heat analysis of Flexible appendages of spacecraft, obtains the transient state temperature field on flexible accessory；
Since the flexibility that high temperature variation causes in the short time is attached when being concerned with spacecraft disengaging earth's shadow emphatically
Part vibration problem, therefore, Orbital heat flux suffered by flexible accessory mainly consider solar radiation hot-fluid.
The heat transfer type that the in-orbit heat analysis of flexible accessory is related to is mainly heat transfer and heat radiation.The heat of in-orbit heat analysis passes
It is identical that fundamental equation, which is led, with the equation of heat conduction under common radiation heat transfer, but increase orbit computation, ascent calculates, is outer
The calculating of hot-fluid.
(3) equivalent heat load leads calculation module, limited using spacecraft Rigid Base-flexible accessory coupled system of foundation
The hot load of transient state temperature field on flexible accessory is equivalent to the joint load on flexible accessory by meta-model；
Using spacecraft Rigid Base-flexible accessory coupled system finite element model of foundation, 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 are generally comprised in flexible accessory structure
Equal components need temperature change suffered on these member units being equivalent to the nodal force load changed over time, so as to
Carry out the coupled system micro-vibration time-histories data analysis of next step.
The equivalent nodal force of temperature load is calculated using the initial strain method in finite element method；According to the deformation of unit
Mode, the temperature load of boom unit are equivalent to axial force, and the temperature load of beam element is equivalent to axial force and torque, plate shell list
The temperature load of member is equivalent to face internal force and face moment of face, completes Equivalent Calculation by the integral in unit；
Assuming that the thermal expansion coefficient of boom unit material is α, Young's modulus E, the cross-sectional area of bar is A, the temperature of boom
Degree variation is T, then the axial equivalent load of boom are as follows:
Pt=∫AEαT dA
Assuming that the thermal expansion coefficient of beam element material is α, Young's modulus E, the cross-sectional area of beam is A, and the temperature of beam becomes
T is turned to, then beam element is other than having axial equivalent load, and there are also following equivalent moment loads:
Mt=∫AEαTy dA
Wherein y is coordinate of the point relative to neutral axis on beam cross section；
Assuming that thickness of shell elements is t, area A, the thermal expansion coefficient of material is α, and the temperature change in face is T, up and down
The temperature difference of surface temperature is Δ T, is linear change along the element thickness direction temperature difference；Then shell unit is in addition to there is the equivalent force in face
[Pt] other than, there are also the equivalent moment [M outside facet]:
Wherein, [Bm] and [Dm] be respectively plane stress problem strain-displacement relation matrix and constitutive matrix,
[Bb] and [Db] it is respectively the Curvature Displacement Relationship matrix and constitutive matrix for being plate bending problem；
The design cycle that the equivalent heat load of time varying temperature field leads calculation module is as follows:
(a) Data of Finite Element Model for reading flexible accessory, obtains nodal information, unit information, material section information；
(b) the transient temperature data on each node of flexible accessory are read；
(c) each moment temperature data is recycled:
Each unit is recycled:
Calculate the equivalent nodal force of each unit；
Nodal force is overlapped by node；
Unit circulation terminates；
Moment circulation terminates.
(4) it is rigid to carry out spacecraft center using the direct Superposition Method of iteration WYD-Ritz vector for high-precision model analysis module
The model analysis of body-flexible accessory coupled system, obtains period and the vibration shape of coupled system, i.e. Method for Solving Generalized Eigenproblem K
Φ=λ M Φ, wherein K and M is respectively stiffness matrix and mass matrix, and λ is characterized value, and Φ is feature vector；
Improved efficiency using multinomial technologies such as grouping shift frequency, mode error convergence criterion, the sparse Fast Direct Methods of cell,
Solving precision and reliability.The scale of solving a problem of eigenvalue problem, can up to 30 to 50 ten thousand freedom degrees on common computer at present
Accurately solve up to several hundred a low side mode.Mode error convergence criterion makes the process of model analysis become steady.Test knot
Fruit shows the precision that mode error bit value indicative error can more reflect that eigenvalue problem calculates.When calculating more multi-modal, mode
Error should be as preferred convergence criterion.
The process of high-precision model analysis module is as follows:
I. it initializes: determining the block wide q of Ritz vector method block and generate step number r；Choose initial vector matrix Q0；Setting is each
Move the maximum number of iterations I of axismax
II. it moves axis: calculating and move axis μ, should try to ensure that it is not characteristic value；It decomposes and moves K- μM of axis stiffness matrix=LDLT；
Sturm sequence is checked
III. iteration ImaxIt is secondary, complete rear steering II
(a) k=0,1 .., r-1 are solvedThen using willTo convergent feature vector and
Q1,Q2,…,QkMake M- orthonomalization, and forms Qk+1；
(b) K is calculated in Q=(Q1,Q2,…,Qr) on projection, K*=QTKQ；
(c) qxr rank Eigenvalue Problem K is solved*Φ*=Φ*Λ*；
(d) new approximate characteristic vector X=Q Φ is formed*；
(e) convergence for pressing mode error judgment characteristic value and feature vector, removes convergent feature vector；
If (f) having reached expected characteristic value number, exit；Otherwise using not converged preceding q vector approximation as just
Beginning vector carries out next iteration.
Model analysis needs the stiffness matrix and mass matrix of input system, and output result is preceding several rank mode knots
Fruit, such as frequency, period, the vibration shape, mode error.
(5) high-precision vibration analysis respond module, analytic solutions and mode superposition method based on Duhamel integral carry out center
Rigid body-flexible accessory coupled system high-precision Response Analysis, obtains the time-histories data result of flexible accessory；
High-precision micro-vibration response analysis module is used for the kinetics equation of solving system, the solution based on Duhamel integral
Analysis solution and mode superposition method are developed, and are not needed access time integration step, can efficiently and accurately be carried out Rigid Base-
The thermotropic micro-vibration response analysis of flexible accessory coupled system.
Governing equations of motion is decoupled using mode superposition method first, obtains one group of non-coupled second order ordinary differential side
Journey；Its solution is integrated to obtain by Duhamel；
The equivalent time-histories load of involved temperature in the present invention is made of the payload values engraved when series of discrete,
In the case where this piecewise linearity or broken line load, Duhamel integral can be in the hope of analytic solutions, without using numerical value product
Point, so as to avoid the select permeability of integration step.
After Modes Decoupling, the groundwork that structural response calculates just is to solve for equation:
The solution of above formula can be integrated to obtain by Duhamel:
Wherein a (t), b (t), h (t) are respectively the solution of following 3 problems:
The concrete form of solution depends on the size of damping ratio ξ；Three kinds of situations can be divided by the size of damping ratio: Asia damping (ξ
<1), critical damping (ξ=1) and overdamp (ξ>1)；
For piecewise linearity or the load function of broken line form；
In order to integrate convenience, a variable related with speed is also introduced:
In the case where Asia damps situation, enableThen non trivial solution is provided by following two formula:
Wherein:
For the situation of critical damping ξ=1, non trivial solution are as follows:
Wherein
w1=-a4+a1
w2=-a7-a5+a2
w3=-a9-2a8-2a6+2a3；
For the situation of overdamp ξ > 1, non trivial solution are as follows:
Wherein
After the above-mentioned modal coordinate solution for obtaining time-histories data, system motive power equation can be obtained using modes superposition
Solution.
(6) post-processing module is to require to extract according to user and show, export relevant calculated result, such as flexible attached
The temperature variation curve of each node, equivalent load change curve, micro-vibration time-histories data curve on part.
The thermotropic micro-vibration of spaceborne flexible accessory provided by the invention responds Simulation Platform, with high pointing accuracy spacecraft
For engineering background, carry out the analysis and simulation research and corresponding software for calculation of the spaceborne large-scale flexible attachment thermal vibration of practical application
Research and development, achieved the purpose that quick, the thermotropic micro-vibration of effective acquisition large-scale flexible component and its responded with celestial body coupled vibrations；
Meanwhile analyzing emulation platform can push spaceborne large-scale flexible component thermal deformation and the engineering of thermotropic micro-vibration simulation analysis technology to answer
It is verified with examination, and drives Related Supporting Technologies progress, for subsequent New Satellite Project R&D accumulation technology basis.
Detailed description of the invention
Fig. 1 is the structural diagram of the present invention；
Fig. 2 is the equivalent thermal force of the bar of flexible accessory structure, beam, film, plate unit；
Fig. 3 is the celestial body schematic diagram with dual sided battery battle array of embodiment；
Fig. 4 is the cell array node side to light temperature cycle variation diagram (4 periods) of embodiment；
Fig. 5 is cell array node side to light and shady face temperature variation (1 period) of embodiment；
Fig. 6 is cell array node side to light and shady face difference variation figure (1 period) of embodiment；
Fig. 7 is side to light and shady face temperature variation curve when the cell array of embodiment enters shade；
Fig. 8 be the cell array of embodiment into shade when side to light and shady face difference variation curve；
Fig. 9 is side to light and shady face temperature variation curve when the cell array of embodiment goes out shade；
Figure 10 is side to light and shady face difference variation curve when the cell array of embodiment goes out shade；
Figure 11 is the sun synchronization of embodiment into ecliptic time section equivalent nodal force change curve；
Figure 12 is the sun synchronization of embodiment into ecliptic time section equivalent node torque change curve；
Figure 13 is that the sun synchronization of embodiment goes out ecliptic time section equivalent nodal force change curve；
Figure 14 is that the sun synchronization of embodiment goes out ecliptic time section equivalent node torque change curve；
Figure 15 is x, y of the outermost side edge mid-points of solar panel of embodiment to displacement (sun synchronization, into ground shadow)；
Figure 16 is the z of the outermost side edge mid-points of solar panel of embodiment to displacement (sun synchronization, into ground shadow)；
Figure 17 is x, y of the outermost side edge mid-points of solar panel of embodiment to displacement (sun synchronization, out shadow)；
Figure 18 is the z of the outermost side edge mid-points of solar panel of embodiment to displacement (sun synchronization, out shadow).
Specific embodiment
Objects and advantages in order to further illustrate the present invention with reference to the accompanying drawing make into one the present invention with concrete case
The explanation of step.
As shown in Figure 1, the spaceborne thermotropic micro-vibration of flexible accessory responds Simulation Platform, including data input modeling mould
Block, in-orbit thermal analysis module, equivalent heat load, which are led, calculates module, high-precision model analysis module, high-precision Response Analysis mould
Block, post-processing module, modules are sequentially connected, input of the output of previous module as next module.
As shown in Fig. 2, generally comprising the components such as boom, beam, plate shell in flexible accessory structure, need these member units
Upper suffered temperature change is equivalent to the nodal force load changed over time, to carry out the coupled system micro-vibration of next step
Time-histories data analysis.
By taking the sun synchronous satellite with dual sided battery battle array as an example, as shown in figure 3, to spacecraft during passing in and out earth's shadow
Since micro-vibration response has carried out Numerical Simulation caused by temperature change.Cell array in the case is by six pieces of solar panels
Composition, between be hinged, cell array and cabin are connected with beam.Classification of track uses sun-synchronous orbit.
Using platform proposed by the present invention, implement numerical simulation according to the following steps:
(1) finite element model and in-orbit thermal model of coupled system are established using data input modeling module.
(2) the in-orbit heat analysis that solar battery array is carried out using in-orbit thermal analysis module passes through the heat analysis such as thermally conductive, radiation
Calculating acquisition Satellite vapour image flexible structure in-orbit period is especially into and out time varying temperature field data when earth's shadow area.
Being 18:00 (into ground shadow) when place, the in-orbit heat analysis meter of satellite sun geo-stationary orbit in 4 periods is made
It calculates, takes the representative node of certain of solar panel side to light and shady face, temperature variations are as shown in Figure 4.Period is
5496.057s, range of temperature are side to light -79.5 to 73.1 DEG C, and shady face -80.2 is to 63.6 DEG C, side to light and backlight
Nearly 10 DEG C of the face temperature difference.
The data of a cycle are taken to analyze, as shown in figure 5, with certain representative node (certain point near solar panel center)
For side to light and shady face temperature: since local time 18:00 (t=0s), solar panel initially enters shadow region, temperature by
It gradually reduces, but is clearly visible that cooling area mid-early stage cooling extent is big, and the later period gradually tends to mitigate；2200s or so reaches side to light
- 79.5 DEG C of lowest temperature (shady face is -80.2 DEG C) left and right, temperature increases suddenly later, initially enters area of illumination, and in 4180s
Left and right reaches 73.1 DEG C of side to light maximum temperature (shady face is 63.6 DEG C).From figure it can also be seen that, when in area of illumination, battery
Plate side to light is obviously higher with respect to shady face temperature.
Fig. 6 is the difference variation figure of the node side to light and shady face.From the results, it was seen that in area of illumination, temperature difference
It is held essentially constant, close to 11.5 DEG C or so；In shadow region, the temperature difference there is no, close to 1 DEG C；But in the two regions
In very short tens seconds in junction, acute variation occurs for the temperature difference.
To in this present embodiment, major concern cell array temperature change in local time's section in disengaging earth's shadow area is drawn
The heat-driven oscillation risen, therefore the time varying temperature field data of following two period is taken to lead calculation to be applied to the equivalent of subsequent temperature loading
And Response Analysis: (1) cell array enters local time's section of shade: (6000s or so satellite is second by 5800s to 7000s
A period initially enters shade), temperature and difference variation on typical node are shown in Fig. 7 and Fig. 8；(2) cell array goes out the office of shade
Portion's period: 7600s to 8000s (7680s or so starts shade out), temperature and difference variation such as Fig. 9 on typical node and
Shown in Figure 10.
(3) calculation module is led using equivalent heat load to be equivalent to be applied to too by the time varying temperature field data that previous step obtains
Hot load on positive cell array flexible structure node；The heat of right side ragged edge solar panel angle point in model is given in Figure 11~14
The case where equivalent result of load nodal force and torque, left side, can be obtained by symmetry.
(4) model analysis is carried out to Coupling System of Flexible Structures And Rigid Body model using high-precision model analysis module；Entire spacecraft is made
For center rigid body-flexible accessory coupled system, three translational degree of freedom of central point are only constrained.The thermal deformation of cell array is in addition to face
Except the bending of outer Z-direction, there are also in-plane deformation X to the deformation with Y-direction, therefore, the vibration shape intercepted when calculating vibration time-histories data
It should include the vibration shape in these three sufficiently more directions.Since the corresponding frequency of the vibration shape is very high into face by X, in order to guarantee response point
The precision of analysis, it is total to have calculated 200 rank mode.Preceding 3 mode is rigid body mode, and the period of the 4th~6 rank mode is respectively as follows: 13.1
Second, 3.87 seconds and 2.30 seconds.
(5) high-precision Response Analysis module is utilized, the time-histories of thermotropic micro-vibration is carried out to Coupling System of Flexible Structures And Rigid Body model
Response analysis.The dynamic respond that the outermost side edge mid-points of solar panel during passing in and out earth's shadow are set forth in Figure 15~18 is bent
Line.
Interpretation of result:
For sun synchronous satellite during passing in and out earth's shadow, the displacement of the lines of X-direction is maximum, followed by the line of Z-direction
The displacement of the lines value of displacement, Y-direction is minimum.Wherein the response of the displacement of the lines of X and Y-direction is quasi-static, and there is no oscillation phenomenons；
The displacement of the lines of Z-direction then has lesser fluctuation, the reason is that sun synchronous satellite is when passing in and out earth's shadow area, thermal response time
(20 seconds or so) are closer to structural cycle (13 seconds or so), to induce thermal vibration.
Claims (4)
1. the thermotropic micro-vibration of spaceborne flexible accessory responds Simulation Platform, it is characterised in that: including data input modeling module,
In-orbit thermal analysis module, equivalent heat load lead calculate module, high-precision model analysis module, high-precision Response Analysis module,
Post-processing module, modules are sequentially connected, input of the output of previous module as next module；
(1) data input modeling module, and it is attached that spacecraft Rigid Base-flexibility is established in the way of interactive mode combination automatic conversion
The finite element model of part coupled system and in-orbit thermal model；
(2) in-orbit thermal analysis module utilizes the in-orbit heat analysis mould of spacecraft Rigid Base-flexible accessory coupled system of foundation
Type 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, utilizes spacecraft Rigid Base-flexible accessory coupled system finite element mould of foundation
The hot load of transient state temperature field on flexible accessory is equivalent to the joint load on flexible accessory by type；
(4) it is soft to carry out spacecraft Rigid Base-using the direct Superposition Method of iteration WYD-Ritz vector for high-precision model analysis module
The model analysis of property attachment coupled system, obtains period and the vibration shape of coupled system, i.e. Method for Solving Generalized Eigenproblem K Φ=λ M
Φ, wherein K and M is respectively stiffness matrix and mass matrix, and λ is characterized value, and Φ is feature vector；
(5) high-precision vibration analysis respond module, it is rigid that analytic solutions and mode superposition method based on Duhamel integral carry out center
Body-flexible accessory coupled system high-precision Response Analysis, obtains the time-histories data result of flexible accessory；
The high-precision vibration analysis respond module first decouples governing equations of motion using mode superposition method, obtains
The second order ordinary differential equation non-coupled to one group；
After Modes Decoupling, solution equation is carried out:
The solution of above formula can be integrated to obtain by Duhamel:
Wherein a (t), b (t), h (t) are respectively the solution of following 3 problems:
The concrete form of solution depends on the size of damping ratio ξ；Three kinds of situations can be divided by the size of damping ratio: ξ < 1 is sub- damping,
ξ=1 is critical damping, and ξ > 1 is overdamp；
For piecewise linearity or the load function of broken line form；
In order to integrate convenience, a variable related with speed is also introduced:
In the case where Asia damps situation, enableThen non trivial solution is provided by following two formula:
Wherein:
For the situation of critical damping ξ=1, non trivial solution are as follows:
Wherein
w1=-a4+a1
w2=-a7-a5+a2
w3=-a9-2a8-2a6+2a3；
For the situation of overdamp ξ > 1, non trivial solution are as follows:
Wherein
After the modal coordinate solution for obtaining time-histories data, system motive power non trivial solution can be obtained using modes superposition；
(6) temperature variation curve, the equivalent load variation song that post-processing module: extracting and shows, exports each node of flexible accessory
Line, time-histories data curve calculated result.
2. the thermotropic micro-vibration of spaceborne flexible accessory according to claim 1 responds Simulation Platform, it is characterised in that: institute
The equivalent heat load stated leads calculation module, and the equivalent node of temperature load is calculated using the initial strain method in finite element method
Power；According to the deformation pattern of unit, the temperature load of boom unit is equivalent to axial force, and the temperature load of beam element is equivalent to axis
To power and torque, the temperature load of Shell Finite Element is equivalent to face internal force and face moment of face, is completed by the integral in unit equivalent
It calculates；
Assuming that the thermal expansion coefficient of boom unit material is α, Young's modulus E, the cross-sectional area of bar is A, and the temperature of boom becomes
T is turned to, then the axial equivalent load of boom are as follows:
Pt=∫AEαT dA
Assuming that the thermal expansion coefficient of beam element material is α, the cross-sectional area of Young's modulus E, beam are A, and the temperature change of beam is
T, then beam element is other than having axial equivalent load, and there are also following equivalent moment loads:
Mt=∫AEαTydA
Wherein y is coordinate of the point relative to neutral axis on beam cross section；
Assuming that thickness of shell elements is t, area A, the thermal expansion coefficient of material is α, and the temperature change in face is T, upper and lower surface
The temperature difference of temperature is Δ T, is linear change along the element thickness direction temperature difference；Then shell unit is in addition to there is the equivalent force [P in facet] with
Outside, there are also the equivalent moment [M outside facet]:
Wherein, [Bm] and [Dm] be respectively plane stress problem strain-displacement relation matrix and constitutive matrix, [Bb] and
[Db] it is respectively the Curvature Displacement Relationship matrix and constitutive matrix for being plate bending problem.
3. the thermotropic micro-vibration of spaceborne flexible accessory according to claim 1 or 2 responds Simulation Platform, feature exists
In: the process that the equivalent heat load leads calculation module is as follows:
(a) Data of Finite Element Model for reading flexible accessory, obtains nodal information, unit information, material section information；
(b) the transient temperature data on each node of flexible accessory are read；
(c) each moment temperature data is recycled:
Each unit is recycled:
Calculate the equivalent nodal force of each unit；
Nodal force is overlapped by node；
Unit circulation terminates；
Moment circulation terminates.
4. the thermotropic micro-vibration of spaceborne flexible accessory according to claim 1 responds Simulation Platform, it is characterised in that: institute
The process for the high-precision model analysis module stated is as follows:
I. it initializes: determining the block wide q of Ritz vector method block and generate step number r；Choose initial vector matrix Q0；Setting moves axis every time
Maximum number of iterations Imax
II. it moves axis: calculating and move axis μ, should try to ensure that it is not characteristic value；It decomposes and moves K- μM of axis stiffness matrix=LDLT；Sturm
Sequence is checked
III. iteration ImaxIt is secondary, complete rear steering II
(a) k=0,1 .., r-1 are solvedThen using willTo convergent feature vector and Q1,
Q2,…,QkMake M- orthonomalization, and forms Qk+1；
(b) K is calculated in Q=(Q1,Q2,…,Qr) on projection, K*=QTKQ；
(c) qxr rank Eigenvalue Problem K is solved*Φ*=Φ*Λ*；
(d) new approximate characteristic vector X=Q Φ is formed*；
(e) convergence for pressing mode error judgment characteristic value and feature vector, removes convergent feature vector；
If (f) having reached expected characteristic value number, exit；Otherwise using not converged preceding q vector approximation as initially to
Amount carries out next iteration.
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