CN104573372B - A kind of netted deployable antenna expansion process Suo Li analysis methods - Google Patents

A kind of netted deployable antenna expansion process Suo Li analysis methods Download PDF

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CN104573372B
CN104573372B CN201510025302.9A CN201510025302A CN104573372B CN 104573372 B CN104573372 B CN 104573372B CN 201510025302 A CN201510025302 A CN 201510025302A CN 104573372 B CN104573372 B CN 104573372B
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CN104573372A (en
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张逸群
茹文锐
段宝岩
杨东武
杜敬利
李申
徐虎荣
杨癸庚
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西安电子科技大学
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Abstract

The invention provides a kind of netted deployable antenna expansion process Suo Li analysis methods, its key step includes:Select netted deployable antenna hoop truss unit number and the discrete operating mode number M of geometric parameter, the material parameter D of rope net, geometric parameter S, rope net topology structure, expansion process;Calculate the movement locus that truss changes with rope net tie point position with the angle of spread;Calculate the stressing conditions of each cable elements under each operating mode;Rope net is fitted to the active force of border junction with the change curve F (θ) of the angle of spread;Flexible multibody dynamics model finally is substituted into using the rope net active force of change as load, carries out dynamic analysis.The present invention regards the expansion process of antenna as the load-bearing balance of mechanism form of series of discrete, study the tension distribution of each equilibrium figure, accurately obtain expansion process Suo Li change curves, modeled and analyzed based on Flexible Multibody System Dynamics, realize that rope net tension force provides effectively support to the Influencing Mechanism analysis for deploying process for ground surface works.

Description

A kind of netted deployable antenna expansion process Suo Li analysis methods

Technical field

The present invention relates to Boundary motion Cable Structure tension analysis and its applied technical field, is specifically a kind of netted deployable Antenna deploys process Suo Li analysis methods.

Background technology

The expansion process of deployable antenna is a complicated nonlinear mechanics process, is from unstable state to stablizing shape State, the conversion from mechanism to structure.In-orbit smoothly expansion is the basis of deployable antenna application, and is easiest to what is broken down One of link.Accurate analysis thus is made to nonlinear influencing factors such as flexibility, friction, rope net tension force in the design phase, realized Prediction to antenna expansion process kineticses behavior, has directive function to the structure design of improve and perfect antenna.Reflecting surface rope Collapsed among truss structure during net transmitting, be gradually tensioned with the expansion of truss structure, ultimately form designed antenna Shape face.In rope net system, Mermis has the characteristics of big displacement small strain, geometrical non-linearity, with antenna exhibition in flexible structure Open, while rope net is progressively tensioned, the nonlinear force of a complicated change can be produced to truss structure, by the exhibition to antenna Open procedure produces the influence that can not ignore.

For rope net during expansion and the mechanical analysis of truss structure interphase interaction, study and often assume before:1> Rope net acts power to truss since being deployed the process latter end moment (at the time of general selection antenna duration of run 80%); 2>Rope tensility is equivalent to spring force, its size is linearly increasing with the expansion of antenna.In this approximation method, rope net tension force pair The active force loading moment of truss is typically set by engineering staff by experience, is theoretically unsound;Simultaneously, it is assumed that rope tensility with The linear change of duration of run, also with being actually deviated, thus ultimately result in analysis result and produce larger error.Also have The method for being modeled and analyzing based on a large amount of cable-truss approximating curve ropes is studied, but either amount of calculation still approaches mistake Difference, all it is difficult to meet engine request.

The content of the invention

The purpose of the present invention is to overcome defect present in above-mentioned prior art, there is provided a kind of netted deployable antenna expansion Process Suo Li analysis methods.For the netted deployable antenna under ground gravity environment, its rope net system is described and built Its mechanical model is found, is resolved and analyzed by mechanics, obtained the equilibrium configuration of any instant (relaxation/tensioning) cable net structure, enter And try to achieve the change curve of the now interaction force between rope net and truss structure.It is finally based on Flexible Multibody System Dynamics Modeling and analysis, realize that rope net tension force is studied the Influencing Mechanism for deploying process.

The technical scheme is that:A kind of netted deployable antenna expansion process Suo Li analysis methods, including following step Suddenly:

Step 101:Select netted deployable antenna hoop truss unit number N, truss element cross bar length L1, truss list First montant length L2, the material parameter D of rope net, geometric parameter S, rope net topology structure, determine the discrete operating mode number of expansion process M;When netted deployable antenna hoop truss unit number is N, then truss and rope net tie point number are 2N;

Step 102:Calculate netted deployable antenna hoop truss and rope net tie point positionWith angle of spread θ={ θj} The movement locus of change, wherein θj∈ [0,90] (j=1,2, M);

Step 103:The rope net tie point position that read step 102 obtainsThe rope net established respectively under Action of Gravity Field hangs Chain line FEM model, obtain the stressing conditions of each cable elements under each operating mode;

Step 104:Based on least square method, active force of the fitting rope net to border junctionWith angle of spread θj's Change curve F (θ);

Step 105:Each border junction is added to using the rope net active force of change as load, substitutes into flexible multi-body dynamics Model is learned, carries out dynamic analysis, you can obtains the displacement of antenna any node, speed during deploying under the influence of rope tensility And acceleration situation.

Above-mentioned steps 102, comprise the following steps:

Step 201:Choose body coordinate system Ox corresponding to any truss element i foundationiyizi, subscript i=1,2, N, N are truss element number;Ai、Bi、Ci、DiRespectively four end points of truss element, θjFor the angle of spread, i.e. cross bar AiDiWith axle xiAngle;Provide Ox1y1z1Body coordinate system overlaps with inertial coodinate system, y in each unitiAxle and bar AiBiOverlap, and unit four Side shape each point is all in OxiyiziIn plane;OxiyiziBody coordinate system and Oxi+1yi+1zi+1The x-axis angle of body coordinate system is

Step 202:From OxiyiziBody coordinate system is to Oxi+1yi+1zi+1The conversion of body coordinate system:The conversion process can be with equivalent For OxiyiziBody coordinate system is first from OiPoint moves to Oi+1Point, i.e. DiAt point, then further around yiTurn over counterclockwiseAngle, convert square Battle array be:

X in formulaDiAnd YDiRespectively DiPoint is in Ox1y1z1Under xiAnd yiCoordinate value;

Step 203:Arbitrfary pointjP is in OxiyiziIn body coordinate system with Oxi+1yi+1zi+1Body coordinate system is respectively depicted asjPi= (jPxi,jPyi,jPzi,1)TWithjPi+1=(jPxi+1,jPyi+1,jPzi+1,1)TAnd the two meets transformation equation:

jPi=iTi+1·jPi+1 (2)

By recurrence formula, during antenna truss expansion, arbitrfary pointjP is from OxiyiziBody coordinate system transformation is to inertia Coordinate system Ox1y1z1On position coordinates be described as:

As

Step 204:WilljPiValue is rope net and the coordinate value at truss element tie point, and purlin can be tried to achieve by formula (4) Frame unit and rope net tie point positionWith angle of spread θjThe movement locus of change, wherein

Above-mentioned steps 103, comprise the following steps:

Step 301:Rope net connects point coordinates under each operating mode that read step 102 obtainsAs boundary node position;

Step 302:According to boundary node positionWith rope hop count X, by Difference Calculation, set and freely saved among rope net Point positionInitial value;

Step 303:Stretched wire clue pessimistic concurrency control under gravity is established, calculates each rope section nodal force in rope net system

Step 304:The node for each rope section that the rope net topology relation and step 303 given in read step 101 obtains PowerNodal force with same node point number is added, node is tried to achieve and makes a concerted effort FClose=[Fx Fy Fz]T, wherein FxFor x side To make a concerted effort, FyFor making a concerted effort for y directions, FzFor making a concerted effort for z directions;

Step 305:Judge the suffered F that makes a concerted effort of any free nodeCloseWhether equilibrium condition is met:FClose=0, such as meet, go to Step 307;If be unsatisfactory for, step 306 is gone to;

Step 306:Make a concerted effort F according to suffered by current structure free nodeCloseWith the stiffness matrix K of slack line net, pass through [K] { Δ u }={ Δ λ } calculates adjustment of displacement amount { Δ u }, wherein { Δ λ } is one a small amount of, and direction and FCloseIt is identical, update free node Position Pfree=Pfree+ { Δ u }, goes to step 303;

Step 307:Export rope net form under each operating mode, the tension force of each cable elementsAnd the stress of border junction

Above-mentioned steps 303, comprise the following steps:

Step 401:Read boundary node positionInformation and the free node position of centreInformation;

Step 402:According to slack line net topology relation, i.e. each unit number and the corresponding relation of two node numbers, one is used Individual catenary elements are described;

Step 403:For arbitrary catenary cable element e, horizontal direction power H under unit coordinate system is obtainede(e=1, 2, X), from following equation, solve to obtain with Newton iteration method:

Wherein,

In formula, A is cross-sectional area, and E is modulus of elasticity, q0For from heavy load, L0Former long for cable elements, l is unit coordinate system Horizontal span, h be unit coordinate system sag;

Step 404:According to gained horizontal direction power He, solve and obtain two-end-point m, n nodal force

In formula, F 'e1、F′e2、F′e3For m points respectively under unit coordinate system x ', y ', z ' directions power, F 'e4、F′e5、F′e6 For n points respectively under unit coordinate system x ', y ', z ' directions power;

Step 405:According to the nodal force F of every rope sectione', the rope net topology structure defined by step 101, group integrates For each rope section nodal force

Above-mentioned steps 104, comprise the following steps:

Step 501:Read 103 and obtain the stressing conditions of each border junction under each operating modeAnd corresponding to operating mode Angle of spread θj

Step 502:Based on the principle of least square with t rank multinomials to either boundary tie point stressing conditionsIt is fitted, that is, forms inconsistent equation group

Step 503:Utilize solution by iterative method solution of equations αb=(αb0b1b2,···,αbt)T, obtain any side The polynomial fitting of the least square data of boundary's tie point:

pb(θ)=αb0b1θ+αb2θ2+···+αbtθt; (9)

Step 504:By the polynomial fitting p of all border junctionsb(θ) group collection, obtains rope net to border junction Active forceWith angle of spread θjChange curve:

F (θ)={ pb(θ) } (b=1,2,2N). (10)

Above-mentioned steps 105, comprise the following steps:

Step 601:The multi-body Dynamics Model of deployable antenna is established based on Lagrangian method, with reference to Rayleigh-inner hereby Deformable body is described method, final to obtain deployable antenna flexible multibody dynamics model:

Wherein M is mass matrix, K be stiffness matrix,It is Lagrange multiplier, Q for Jacobian matrix, λFFor broad sense master Power, QvIt is the generalized coordinates vector chosen for related generalized force secondary to speed, q;

Step 602:Each border junction stress curve obtained by step 104 is loaded onto truss corresponding positions in the form of external force Put, i.e., kinetics equation is changed into

Step 603:Equation (12) is solved based on Newmark methods, that is, obtains displacement, the speed of antenna any node Degree and acceleration situation.

Beneficial effects of the present invention:The present invention by the expansion process of antenna by being converted into multiple transient buildups, for every One particular state, the rope net system that can be attributed to a known rope Duan Yuanchang and boundary point position look for shape problem, while base In catenary elements, establish the FEM model of the net-shape antenna rope net under gravity environment, can accurately obtain each rope section form with Tension force situation.The present invention can accurately obtain the change curve of netted deployable antenna expansion process Suo Li changes, based on flexibility Dynamics of multibody systems models and analysis, realizes that rope net tension force is provided with to the Influencing Mechanism analysis for deploying process for ground surface works Effect support.Its committed step is exactly to move independently of form variable, i.e., regards motion process as load-bearing by series of discrete Balance of mechanism form forms, and then studies the tension distribution of each equilibrium figure.It is an advantage of the invention that:1) can be accurately obtained Rope net acts the change curve of power during deployable antenna expansion;2) obtained rope net active force change curve is substituted into Flexible multibody dynamics model, impact analysis of the rope net tension force to expansion process can be carried out, be deployable antenna motor and control System design provides to be instructed in advance, avoids expansion process from deploying not in place or wild effect.

The present invention is described in further details below with reference to accompanying drawing.

Brief description of the drawings

Fig. 1 unfolded reticular antenna schematic diagrames;

Fig. 2 truss elements coordinate system description figure;

The main flow chart of the netted deployable antenna expansion process Suo Li analysis methods of Fig. 3;

Fig. 4 calculates the movement locus process that rope net changes with truss tie point with the angle of spread;

Fig. 5 calculates each cable elements stressing conditions process of rope net under each operating mode;

Fig. 6 establishes rope pessimistic concurrency control and calculates each rope section nodal force process in rope net system;

The process that Fig. 7 fitting rope nets change to border junction active force with the angle of spread;

Fig. 8 rope net tie points stressing conditions substitute into kinetic model solution process;

Fig. 9 the inventive method is applied to the rope net active force variation diagram emulated on certain deployable rope net antenna structure;

Figure 10 the inventive method is applied to the dynamic analysis driving force emulated on certain deployable rope net antenna structure Variation diagram;

Description of reference numerals:1st, montant;2nd, cross bar;3rd, free node;4th, rope net border junction.

Embodiment

As shown in figure 3, the invention provides a kind of unfolded reticular antenna to deploy process Suo Li analysis methods, including it is as follows Step:

Step 101:Referring to Fig. 1, selecting netted deployable antenna hoop truss unit number N, (then truss is connected with rope net Point number be 2N), truss element cross bar 2 and the length L of montant 11And L2, material parameter D, geometric parameter S, the rope net topology of rope net Structure, determine the discrete operating mode number M of expansion process;

Step 102:Calculate netted deployable antenna hoop truss and rope net tie point positionWith angle of spread θ={ θj} The movement locus of change, wherein θj∈ [0,90] (j=1,2, M);As shown in figure 4, this step comprises the following steps:

Step 201:Referring to Fig. 2, body coordinate system Ox corresponding to any truss element i foundation is choseniyizi, subscript i=1, 2, N, N are truss element number.Ai、Bi、Ci、DiRespectively four end points of truss element, θjIt is for the angle of spread, i.e., horizontal Bar AiDiWith axle xiAngle.Provide Ox1y1z1Body coordinate system overlaps with inertial coodinate system, y in each unitiAxle and bar AiBiOverlap, And unit quadrangle each point is all in OxiyiziIn plane.OxiyiziBody coordinate system and Oxi+1yi+1zi+1The x-axis angle of body coordinate system For

Step 202:From OxiyiziBody coordinate system is to Oxi+1yi+1zi+1The conversion of body coordinate system.The process can be equivalent to handle OxiyiziBody coordinate system is first from OiPoint moves to Oi+1Point (i.e. DiPoint) place, then further around yiTurn over counterclockwiseAngle.Transformation matrix For:

X in formulaDiAnd YDiRespectively DiPoint is in Ox1y1z1Under xiAnd yiCoordinate value.

Step 203:Arbitrfary pointjP is in OxiyiziIn body coordinate system with Oxi+1yi+1zi+1Body coordinate system can be respectively depicted asjPi =(jPxi,jPyi,jPzi,1)TWithjPi+1=(jPxi+1,jPyi+1,jPzi+1,1)TAnd the two meets transformation equation:

jPi=iTi+1·jPi+1 (2)

By recurrence formula, during antenna truss expansion, arbitrfary pointjP is from OxiyiziBody coordinate system transformation is to inertia Coordinate system Ox1y1z1On position coordinates can be described as:

As

Step 204:WilljPiValue is rope net and the coordinate value at truss element tie point, and purlin can be tried to achieve by formula (4) Frame unit and rope net tie point positionWith angle of spread θjThe movement locus of change, wherein

Step 103:The position of rope net tie point 4 that read step 102 obtainsThe rope net established respectively under Action of Gravity Field Catenary FEM model, obtain the stressing conditions of each cable elements under each operating mode;As shown in figure 5, this step includes following step Suddenly:

Step 301:Rope net connects point coordinates under each operating mode that read step 102 obtainsAs boundary node position;

Step 302:According to boundary node positionWith rope hop count X, by Difference Calculation, set and freely saved among rope net 3 positions of pointInitial value;

Step 303:Stretched wire clue pessimistic concurrency control under gravity is established, calculates each rope section nodal force in rope net systemAs shown in fig. 6, this step 303 comprises the following steps:

Step 401:Read boundary node positionInformation and the free node position of centreInformation;

Step 402:According to slack line net topology relation, i.e. each unit number and the corresponding relation of two node numbers, one is used Individual catenary elements are described;

Step 403:For arbitrary catenary cable element e, horizontal direction power H under unit coordinate system is obtainede(e=1, 2, X), from following equation, solve to obtain with Newton iteration method:

Wherein,

In formula, A is cross-sectional area, and E is modulus of elasticity, q0For from heavy load, L0Former long for cable elements, l is unit coordinate system Horizontal span, h be unit coordinate system sag;

Step 404:According to gained horizontal direction power He, solve and obtain two-end-point m, n nodal force

In formula, F 'e1、F′e2、F′e3For m points respectively under unit coordinate system x ', y ', z ' directions power, F 'e4、F′e5、F′e6 For n points respectively under unit coordinate system x ', y ', z ' directions power.

Step 405:According to the nodal force F of every rope sectione', the rope net topology structure defined by step 101, group integrates For each rope section nodal force

Step 304:The node for each rope section that the rope net topology relation and step 303 given in read step 101 obtains PowerNodal force with same node point number is added, node is tried to achieve and makes a concerted effort FClose=[Fx Fy Fz]T, wherein FxFor x side To make a concerted effort, FyFor making a concerted effort for y directions, FzFor making a concerted effort for z directions;

Step 305:Judge the suffered F that makes a concerted effort of any free node 3CloseWhether equilibrium condition is met:FClose=0, such as meet, go to Step 307;If be unsatisfactory for, step 306 is gone to;

Step 306:Make a concerted effort F according to suffered by current structure free node 3CloseWith the stiffness matrix K of slack line net, pass through [K] { Δ u }={ Δ λ } calculates adjustment of displacement amount { Δ u }, wherein { Δ λ } is one a small amount of, and direction and FCloseIt is identical, update free node Position Pfree=Pfree+ { Δ u }, goes to step 303;

Step 307:Export rope net form under each operating mode, the tension force of each cable elementsAnd the stress of border junction

Step 104:Based on least square method, active force of the fitting rope net to border junctionWith angle of spread θj's Change curve F (θ);As shown in fig. 7, this step comprises the following steps:

Step 501:Read 103 and obtain the stressing conditions of each border junction under each operating modeAnd corresponding to operating mode Angle of spread θj

Step 502:Based on the principle of least square with t rank multinomials to either boundary tie point stressing conditionsIt is fitted, that is, forms inconsistent equation group

Step 503:Utilize solution by iterative method solution of equations αb=(αb0b1b2,···,αbt)T, obtain any side The polynomial fitting of the least square data of boundary's tie point:

pb(θ)=αb0b1θ+αb2θ2+···+αbtθt (9)

Step 504:By the polynomial fitting p of all border junctionsb(θ) group collection, obtains rope net to border junction Active forceWith angle of spread θjChange curve:

F (θ)={ pb(θ) } (b=1,2,2N). (10)

Step 105:Each border junction is added to using the rope net active force of change as load, substitutes into flexible multi-body dynamics Model is learned, carries out dynamic analysis, you can obtains the displacement of antenna any node, speed during deploying under the influence of rope tensility And acceleration situation.As shown in figure 8, this step comprises the following steps:

Step 601:The multi-body Dynamics Model of deployable antenna is established based on Lagrangian method, with reference to Rayleigh-inner hereby Deformable body is described method, final to obtain deployable antenna flexible multibody dynamics model:

Wherein M is mass matrix, K be stiffness matrix,It is Lagrange multiplier, Q for Jacobian matrix, λFFor broad sense master Power (square), QvIt is the generalized coordinates vector chosen for related generalized force (square) secondary to speed, q;

Step 602:Each border junction stress curve obtained by step 104 is loaded onto truss corresponding positions in the form of external force Put, i.e., kinetics equation is changed into

Step 603:The differential equation (12) is solved based on Newmark methods, you can obtain antenna any node Displacement, speed and acceleration situation.

The effect of the present invention is verified by following emulation experiment.

Shape method is looked for be applied to be emulated in certain 2 meters of bore rope net deployable antenna structure the present invention, such as Fig. 1 institutes Show.Wherein netted deployable antenna hoop truss unit number N=6, truss element cross bar L1=1m, montant L2=0.6m, own Rope section uses aramid fiber material, and modulus of elasticity is E=2 × 1010Pa, cross-sectional area are A=π/4 × 10-6m3, cable elements number is 97, the discrete operating mode number M=50 of expansion process.

Fig. 9 give the cable net structure being calculated by analysis method of the present invention respectively to upper and lower truss five to joint and The Suo Li situations of change of three-dimensional joint.Understand that antenna five is first subjected to rope net tension force to joint and influenceed when being deployed into 70 °, be deployed into 73 ° or so three-dimensional joints just start stress, thus show deployable antenna overall structure discontinuity equalization, and antenna will be caused to deploy Suffered driving force produces fluctuation.Figure 10 provides substitutes into flexible multibody dynamics model by the rope net active force of change, and analysis obtains Driving force change curve.More meet reality compared with original method assumed based on engineering experience, and accuracy is higher.

To sum up, the present invention by the expansion process of antenna by being converted into multiple transient buildups, for each particular state, The rope net system of a known rope Duan Yuanchang and boundary point position can be attributed to looks for shape problem, while is based on catenary elements, The FEM model of the net-shape antenna rope net under gravity environment is established, can accurately obtain each rope section form and tension force situation.This hair The bright change curve that can accurately obtain netted deployable antenna expansion process Suo Li changes, based on Flexible Multibody System Dynamics Modeling and analysis, realize that rope net tension force provides effectively support to the Influencing Mechanism analysis for deploying process for ground surface works.It is crucial Step is exactly to move independently of form variable, i.e., regards motion process as load-bearing balance of mechanism form group by series of discrete Into then studying the tension distribution of each equilibrium figure.It is an advantage of the invention that:1) deployable antenna expansion can be accurately obtained During rope net act the change curve of power;2) obtained rope net active force change curve is substituted into flexible multibody dynamics Model, impact analysis of the rope net tension force to expansion process can be carried out, is provided in advance for deployable antenna motor and Control System Design First instruct, avoid expansion process from deploying not in place or wild effect.

There is no the known conventional means of the part category industry described in detail in present embodiment, do not chat one by one here State.It is exemplified as above be only to the present invention for example, do not form the limitation to protection scope of the present invention, it is every with this Same or analogous design is invented to belong within protection scope of the present invention.

Claims (5)

1. a kind of netted deployable antenna expansion process Suo Li analysis methods, it is characterized in that:Comprise the following steps:
Step 101:Select netted deployable antenna hoop truss unit number N, the length L of truss element cross bar 21, truss element erects The length L of bar 12, the material parameter D of rope net, geometric parameter S, rope net topology structure, determine the discrete operating mode number M of expansion process;When When netted deployable antenna hoop truss unit number is N, then truss and rope net tie point number are 2N;
Step 102:Calculate netted deployable antenna hoop truss and rope net tie point positionWith angle of spread θ={ θjChange Movement locus, wherein θj∈ [0,90], j=1,2, M;
Step 103:The rope net tie point position that read step 102 obtainsThe rope net catenary established respectively under Action of Gravity Field FEM model, obtain the stressing conditions of each cable elements under each operating mode;
Step 104:Based on least square method, active force of the fitting rope net to border junctionWith angle of spread θjChange Curve F (θ);
Step 105:Each border junction is added to using the rope net active force of change as load, substitutes into flexible multibody dynamics mould Type, carry out dynamic analysis, you can obtain the displacement of antenna any node, speed during deploying under the influence of rope tensility and add Speed conditions;
The step 103, comprises the following steps:
Step 301:Rope net tie point position under each operating mode that read step 102 obtainsAs boundary node position;
Step 302:According to boundary node position and rope hop count X, by Difference Calculation, free node position among rope net is setInitial value;
Step 303:Stretched wire clue pessimistic concurrency control under gravity is established, calculates each rope section nodal force in rope net system
Step 304:The nodal force for each rope section that the rope net topology relation and step 303 given in read step 101 obtainsNodal force with same node point number is added, node is tried to achieve and makes a concerted effort FClose=[Fx Fy Fz]T, wherein FxFor x directions Make a concerted effort, FyFor making a concerted effort for y directions, FzFor making a concerted effort for z directions;
Step 305:Judge the suffered F that makes a concerted effort of any free nodeCloseWhether equilibrium condition is met:FClose=0, such as meet, go to step 307;If be unsatisfactory for, step 306 is gone to;
Step 306:Make a concerted effort F according to suffered by current structure free nodeCloseWith the stiffness matrix K of slack line net, pass through [K] { Δ u } ={ Δ λ } calculates adjustment of displacement amount { Δ u }, wherein { Δ λ } is one a small amount of, and direction and FCloseIt is identical, renewal free node position Pfree=Pfree+ { Δ u }, goes to step 303;
Step 307:Export rope net form under each operating mode, the tension force of each cable elementsAnd the stress of border junction
2. a kind of netted deployable antenna expansion process Suo Li analysis methods as claimed in claim 1, it is characterized in that:The step Rapid 102, comprise the following steps:
Step 201:Choose body coordinate system Ox corresponding to any truss element i foundationiyizi, subscript i=1,2, N, N are Truss element number;Ai、Bi、Ci、DiRespectively four end points of truss element, θjFor the angle of spread, i.e. cross bar AiDiWith axle xiFolder Angle;Provide Ox1y1z1Body coordinate system overlaps with inertial coodinate system, y in each unitiAxle and bar AiBiOverlap, and unit quadrangle Each point is all in OxiyiziIn plane;OxiyiziBody coordinate system and Oxi+1yi+1zi+1The x-axis angle of body coordinate system is
Step 202:From OxiyiziBody coordinate system is to Oxi+1yi+1zi+1The conversion of body coordinate system:The conversion process can be equivalent to handle OxiyiziBody coordinate system is first from OiPoint moves to Oi+1Point, i.e. DiAt point, then further around yiTurn over counterclockwiseAngle, transformation matrix are:
X in formulaDiAnd YDiRespectively DiPoint is in Ox1y1z1Under xiAnd yiCoordinate value;
Step 203:Arbitrfary pointjP is in OxiyiziIn body coordinate system with Oxi+1yi+1zi+1Body coordinate system is respectively depicted asjPi=(jPxi ,jPyi,jPzi,1)TWithjPi+1=(jPxi+1,jPyi+1,jPzi+1,1)TAnd the two meets transformation equation:
jPi=iTi+1·jPi+1 (2)
By recurrence formula, during antenna truss expansion, arbitrfary pointjP is from OxiyiziBody coordinate system transformation is to inertial coodinate system Ox1y1z1On position coordinates be described as:
<mrow> <mmultiscripts> <mi>P</mi> <mn>1</mn> <mi>i</mi> <mi>j</mi> </mmultiscripts> <mo>=</mo> <msub> <mmultiscripts> <mi>T</mi> <mn>1</mn> </mmultiscripts> <mi>i</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mmultiscripts> <mi>P</mi> <mi>j</mi> </mmultiscripts> <mi>i</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <munderover> <mo>&amp;Pi;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>i</mi> </munderover> <msub> <mmultiscripts> <mi>T</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </mmultiscripts> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msub> <mmultiscripts> <mi>P</mi> <mi>j</mi> </mmultiscripts> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
As
Step 204:WilljPiValue is rope net and the coordinate value at truss element tie point, and truss list can be tried to achieve by formula (4) Member and rope net tie point positionWith angle of spread θjThe movement locus of change, wherein
<mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>P</mi> <mn>0</mn> <msub> <mi>&amp;theta;</mi> <mi>j</mi> </msub> </msubsup> <mo>=</mo> <mo>{</mo> <mmultiscripts> <mi>P</mi> <mn>1</mn> <mi>i</mi> <mi>j</mi> </mmultiscripts> <mo>}</mo> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>N</mi> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>M</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
3. a kind of netted deployable antenna expansion process Suo Li analysis methods as claimed in claim 1, it is characterized in that:The step Rapid 303, comprise the following steps:
Step 401:Read boundary node positionInformation and the free node position of centreInformation;
Step 402:It is outstanding using one according to slack line net topology relation, i.e. each unit number and the corresponding relation of two node numbers Chain line unit is described;
Step 403:For arbitrary catenary cable element e, horizontal direction power H under unit coordinate system is obtainede, e=1,2, X, from following equation, solve to obtain with Newton iteration method:
<mrow> <mfrac> <mrow> <mn>4</mn> <msup> <msub> <mi>H</mi> <mi>e</mi> </msub> <mn>2</mn> </msup> </mrow> <msubsup> <mi>q</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mfrac> <msup> <mi>sinh</mi> <mn>2</mn> </msup> <mi>&amp;mu;</mi> <mo>+</mo> <mfrac> <msup> <mi>h</mi> <mn>2</mn> </msup> <msup> <mrow> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>+</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>L</mi> <mn>0</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>A</mi> <mi>E</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>coth</mi> <mi>&amp;mu;</mi> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <msubsup> <mi>L</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
Wherein,
In formula, A is cross-sectional area, and E is modulus of elasticity, q0For from heavy load, L0Former long for cable elements, l is the water of unit coordinate system Degree of flatting across, h are the sag of unit coordinate system;
Step 404:According to gained horizontal direction power He, solve and obtain two-end-point m, n nodal force
<mrow> <msubsup> <mi>F</mi> <mi>e</mi> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msubsup> <mi>F</mi> <mrow> <mi>e</mi> <mn>1</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>F</mi> <mrow> <mi>e</mi> <mn>2</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>F</mi> <mrow> <mi>e</mi> <mn>3</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>F</mi> <mrow> <mi>e</mi> <mn>4</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>F</mi> <mrow> <mi>e</mi> <mn>5</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>F</mi> <mrow> <mi>e</mi> <mn>6</mn> </mrow> <mo>&amp;prime;</mo> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mo>-</mo> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mi>H</mi> <mi> </mi> <mi>sinh</mi> <mo>&amp;lsqb;</mo> <msup> <mi>cosh</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mfrac> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>L</mi> <mn>0</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>H</mi> <mi> </mi> <mi>sinh</mi> <mi>&amp;mu;</mi> </mrow> </mfrac> <mo>)</mo> <mo>-</mo> <mi>&amp;mu;</mi> <mo>&amp;rsqb;</mo> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> <mi> </mi> <mi>sinh</mi> <mo>&amp;lsqb;</mo> <msup> <mi>cosh</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mfrac> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>L</mi> <mn>0</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>H</mi> <mi> </mi> <mi>sinh</mi> <mi>&amp;mu;</mi> </mrow> </mfrac> <mo>)</mo> <mo>+</mo> <mi>&amp;mu;</mi> <mo>&amp;rsqb;</mo> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
In formula, F 'e1、F′e2、F′e3For m points respectively under unit coordinate system x ', y ', z ' directions power, F 'e4、F′e5、F′e6For n Put x ', y ', the power in z ' directions under unit coordinate system respectively;
Step 405:According to the nodal force F of every rope sectione', the rope net topology structure defined by step 101, group is integrated into each rope Section nodal force
4. a kind of netted deployable antenna expansion process Suo Li analysis methods as claimed in claim 1, it is characterized in that:The step Rapid 104, comprise the following steps:
Step 501:Read 103 and obtain the stressing conditions of each border junction under each operating modeAnd deploy corresponding to operating mode Angle θj
Step 502:Based on the principle of least square with t rank multinomials to either boundary tie point stressing conditionsB=1, 2,2N is fitted, that is, forms inconsistent equation group
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </msub> <msup> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mi>t</mi> </mrow> </msub> <msup> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>t</mi> </msup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow> <mi>j</mi> <mi>o</mi> <mi>int</mi> <mo>,</mo> <mi>b</mi> </mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </msub> <msup> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mi>t</mi> </mrow> </msub> <msup> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mi>t</mi> </msup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow> <mi>j</mi> <mi>o</mi> <mi>int</mi> <mo>,</mo> <mi>b</mi> </mrow> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>&amp;theta;</mi> <mi>M</mi> </msub> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </msub> <msup> <msub> <mi>&amp;theta;</mi> <mi>M</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>b</mi> <mi>t</mi> </mrow> </msub> <msup> <msub> <mi>&amp;theta;</mi> <mi>M</mi> </msub> <mi>t</mi> </msup> <mo>=</mo> <msubsup> <mi>F</mi> <mrow> <mi>j</mi> <mi>o</mi> <mi>int</mi> <mo>,</mo> <mi>b</mi> </mrow> <msub> <mi>&amp;theta;</mi> <mi>M</mi> </msub> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Step 503:Utilize solution by iterative method solution of equations αb=(αb0b1b2,···,αbt)T, obtain either boundary company The polynomial fitting of the least square data of contact:
pb(θ)=αb0b1θ+αb2θ2+···+αbtθt(9);
Step 504:By the polynomial fitting p of all border junctionsb(θ) group collection, obtains active force of the rope net to border junctionWith angle of spread θjChange curve:
F (θ)={ pb(θ) } (b=1,2,2N) (10).
5. a kind of netted deployable antenna expansion process Suo Li analysis methods as claimed in claim 1, it is characterized in that:The step Rapid 105, comprise the following steps:
Step 601:The multi-body Dynamics Model of deployable antenna is established based on Lagrangian method, with reference to Rayleigh-Ritz theory pair Deformable body is described, final to obtain deployable antenna flexible multibody dynamics model:
<mrow> <mi>M</mi> <mover> <mi>q</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>K</mi> <mi>q</mi> <mo>+</mo> <msubsup> <mi>C</mi> <mi>q</mi> <mi>T</mi> </msubsup> <mi>&amp;lambda;</mi> <mo>=</mo> <msub> <mi>Q</mi> <mi>F</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mi>v</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Wherein M is mass matrix, K be stiffness matrix,It is Lagrange multiplier, Q for Jacobian matrix, λFFor broad sense actively Power, QvIt is the generalized coordinates vector chosen for related generalized force secondary to speed, q;
Step 602:Each border junction stress curve obtained by step 104 is loaded onto truss relevant position in the form of external force, I.e. kinetics equation is changed into
<mrow> <mi>M</mi> <mover> <mi>q</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>K</mi> <mi>q</mi> <mo>+</mo> <msubsup> <mi>C</mi> <mi>q</mi> <mi>T</mi> </msubsup> <mi>&amp;lambda;</mi> <mo>=</mo> <msub> <mi>Q</mi> <mi>F</mi> </msub> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Q</mi> <mi>v</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
Step 603:Equation (12) is solved based on Newmark methods, that is, obtain the displacement of antenna any node, speed and Acceleration situation.
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