CN105912781A - Method for kinetic analysis of antenna deploying involving cable net structure - Google Patents

Abstract

The invention provides a method for kinetic analysis of antenna deploying involving a cable net structure. The method comprises the steps that (1) material parameters and geometric parameters of an antenna truss unit and a cable net as well as a cable net topologic structure and cable net node initial positions are selected; (2) a kinetic model of the cable net is established according to a Second Lagrange Equation; (3) a kinetic model of an antenna truss is established; (4) a constraint equation of the truss unit and the cable net is established, and an overall kinetic model of the antenna is established; and (5) the kinetic models are solved, so that a motion form and cable net acting force are obtained. The method provided by the invention has the advantages that the antenna deploying course involving the cable net structure can be analyzed accurately, so that dynamic changes of a cable net form can be obtained; and a changing curve about the cable net acting force during the deploying course can be obtained accurately, effects of cable net tension nonlinear factors on antenna deploying can be analyzed, a basis can be provided for design of a deployable antenna motor and a control system, and unsteady or inaccurate antenna deploying during the deploying course can be avoided.

Description

A kind of antenna Deployment Dynamic Analysis method containing cable net structure

Technical field

The present invention relates to net-shape antenna and launch process kinetics analysis and applied technical field, particularly expandable mesh Antenna launches the dynamic analysis method of process rope net and truss combining structure, and a kind of antenna containing cable net structure launches Dynamic analysis method.

Background technology

The expansion process of large-scale spaceborne netted deployable antenna is a complicated nonlinear mechanics process, and flexible cable net is opened Between pull-up structure folds and draws in the structure along with truss in rocket launching transportation, arrive at after designed path with truss progressively It is expanded to designed dwi hastasana face.Cable mesh reflector experienced by from the state being relaxed to tensioning in whole expansion process, and by In its flexible strongly geometrically nonlinear feature, it will antenna entirety is produced a complicated nonlinear force, causes complexity many The nonlinear kinetics response become.

Antenna structure containing rope net launches process, and rope net should also be a dynamic complicated mistake in form and mechanics Journey.Due to the lighter weight of cable net structure, the inertia that in the middle of the research of forefathers, major part produces during assuming that rope net unfolding Power is negligible, only considered the active force that during being launched by antenna, antenna is produced by rope net and is applied to purlin as external applied load Truss flexible multibody dynamics analysis is carried out on frame.But, to more accurately analyze the expansion dynamic response of net-shape antenna, should Sky clue net and truss structure are combined modeling, take into full account the dynamics of rope net.

Accordingly, it would be desirable to from the angle of energy, based on elastic catenary unit, utilize Lagrangian second equation to build The kinetic model of rope net, and it is combined modeling with Flexible Truss.Analyzed by Dynamic solving, obtain expansion process The motion morphology of middle rope net and to dynamic responses such as the active forces that Truss joint produces.

Summary of the invention

It is an object of the invention to overcome above-mentioned problems of the prior art, it is provided that a kind of antenna exhibition containing cable net structure Start mechanical analyzing method.Based on elastic catenary model of element, derive the kinetics equation of cable net structure, and derive soft The kinetic model of property truss Cable Structure, obtains antenna integral power model with cable net structure compositional modeling.Thus accurate Antenna under jointly acting on to flexible truss and non-thread sex cords net launches dynamic response, launches with improving antenna for engineering design The planning control of process etc. provide and effectively support.

The technical scheme is that a kind of antenna Deployment Dynamic Analysis method containing cable net structure is characterized in that: bag Include following steps:

Step 101: select netted deployable antenna truss element and the material parameter of rope net, geometric parameter, rope net topology Structure, the initial position P of rope net node；

Step 102: according to Lagrangian second equation, builds the kinetic model of rope net:

$\frac{d}{dt}\left(\frac{\∂T_E}{\∂\stackrel{\·}{q}}\right)-\frac{\∂T_E}{\∂q}+\frac{\∂U_E+U_g}{\∂q}=Q---\left(1\right)$

In formula, q,For the generalized coordinates chosen and the generalized velocity of correspondence thereof, t is the time, T_EFor the kinetic energy of system, U_E For the elastic potential energy of system, U_gFor the gravitional force of system, Q is the generalized force that nonconservative force is corresponding；

Step 103: utilize Ruili-Ritz method to carry out discrete to truss element, the kinetic model of derivation flexible truss；

Step 104: build the constraint equation of truss element and rope net, the kinetic model of rope net Yu truss is carried out group Close；

Step 105: this model is solved based on Newmark method, i.e. can get the motion process of rope net form with The active force situation of change that truss is produced by rope net.

Above-mentioned step 102 specifically includes following steps:

Step 201: be can get arbitrary cable elements i by rope net topology relation, its two node is j and k, the position of its interior joint j Putting coordinate is P_j=[x_j y_j z_j]^T, the position coordinates of node k is P_k=[x_k y_k z_k]^T；The generalized coordinates of cable elements i is described For:

q_i=[x_jy_jz_jx_ky_kz_k]^T (2)

Corresponding generalized velocity is:

${\stackrel{\·}{q}}_i={\[{\stackrel{\·}{x}}_j{\stackrel{\·}{y}}_j{\stackrel{\·}{z}}_j{\stackrel{\·}{x}}_k{\stackrel{\·}{y}}_k{\stackrel{\·}{z}}_k\]}^T---\left(3\right)$

The generalized coordinates then describing whole rope net system is:

Q=[x_ny_nz_n]^T, n=1,2 ..., N (4)

The generalized velocity describing whole rope net system is:

$\stackrel{\·}{q}={\[{\stackrel{\·}{x}}_n{\stackrel{\·}{y}}_n{\stackrel{\·}{z}}_n\]}^T,n=1,2,\mathrm{...},N---\left(5\right)$

Wherein N is rope net node total number；T is matrix transposition symbol；

Step 202: by the uniform mass M of cable elements i_iIt is equivalent to two-end-point lumped mass have:

$M_j=M_k=\frac{1}{2}{M}_i---\left(6\right)$

If the element number being connected with any node k is c_i, the unit sum being connected is c_N, then the kinetic energy T at node k_k For:

$T_k=\frac{1}{4}({{\stackrel{\·}{x}}_k}^2+{{\stackrel{\·}{y}}_k}^2+{{\stackrel{\·}{z}}_k}^2)\underset{i=1}{\overset{c_N}{\Σ}}{M}_{ci}---\left(7\right)$

The then kinetic energy T of rope net system_cFor:

$T_c=\underset{k=1}{\overset{N}{\Σ}}{T}_k---\left(8\right)$

Step 203: by comparing the relation of cable elements chord length and former length, it is judged that whether cable elements is in tensioning state；If Cable elements chord length is more than former length, cable elements tensioning, forwards step 204 to；If cable elements chord length is less than or equal to former length, cable elements pine Relax, forward step 205 to；

Step 204: cable elements chord length is more than former length, and cable elements tensioning, its mechanical stae meets Hooke theorem；Wherein jk ' For state before cable elements generation elastic deformation, jk is the tensioning state after catenary elements stress；Now, cable elements elastic potential energy U_EiCan be reduced to:

$U_{Ei}=\frac{EA}{2L_{0i}}{(L_{jk}-L_{0i})}^2---\left(9\right)$

L_jk ²=(x_j-x_k)²+(y_j-y_k)²+(z_j-z_k)² (10)

Wherein E is elastic modelling quantity, and A is cable elements sectional area；

Unit gravitional force U_giFor barycenter potential energy:

$U_{gi}=M_{i}g\frac{z_j+z_k}{2}---\left(11\right)$

In formula, g is acceleration of gravity.

Pass directly to step 206；

Step 205: cable elements chord length is less than or equal to former length, cable element slack, is derived by slack line list by unit The elastic potential energy of unit and gravitional force；

Step 206: the elastic potential energy of rope net system is:

$U_E=\underset{i=1}{\overset{N_e}{\Σ}}{U}_{Ei}---\left(12\right)$

The gravitional force of rope net system is:

$U_g=\underset{i=1}{\overset{N_e}{\Σ}}{U}_{gi}---\left(13\right)$

Wherein, N_eFor cable elements sum；

Step 207: rope net generalized coordinates, kinetic energy, elastic potential energy, gravitional force are substituted into formula (1), obtains the dynamic of rope net Mechanical model.

Above-mentioned step 205, specifically includes following steps:

Step 301: lax catenary elements is by uniform load q₀Time, unit is sagging trend；Point S is some S₀There is bullet Property deformation after corresponding position；T is the tension force on cable elements at any point S, a length of s of point from starting point to S, corresponding former a length of s₀；Whole machine balancing equation both horizontally and vertically all meets for any point S:

$T\frac{dx}{ds}=H---\left(14\right)$

$T\frac{dz}{ds}=q_0{s}_0-F_3^{\′}---\left(15\right)$

Step 302: meet geometrical constraint at any point S:

${\left(\frac{dx}{ds}\right)}^2+{\left(\frac{dz}{ds}\right)}^2=1---\left(16\right)$

Assume that material meets Hooke's law:

$T=AE(\frac{ds}{{\mathrm{ds}}_0}-1)---\left(17\right)$

Equation (16) will be substituted into after equation (14) and (15) summed square, can obtain the rope tensility on any point S point:

$T\left(s_0\right)={\[H^2+{(q_0{s}_0-F_3^{\′})}^2\]}^{1/2}---\left(18\right)$

Wherein H, F₃' by the fundamental equation of unit, obtained by numerical solution；

Step 303: elastic potential energy dU on any point S can be obtained by strain energy formulation_E:

${\mathrm{dU}}_E=\frac{T{\left(s_0\right)}^2{\mathrm{ds}}_0}{2EA}---\left(19\right)$

Thus the elastic potential energy of cable elements ab is:

$U_{Ei}={\∫}_0^{L_{0i}}{\mathrm{dU}}_E={\∫}_0^{L_{0i}}\frac{T{\left(s_0\right)}^2}{2EA}{\mathrm{ds}}_0---\left(20\right)$

Step 304: by conversion dz/ds=(dz/ds₀)(ds₀/ ds), formula (15) and formula (17) obtain catenary Z is to coordinate:

$z\left(s_0\right)=\frac{F_3^{\′}{s}_0}{AE}(\frac{q_0{s}_0}{2F_3^{\′}}-1)+\frac{H}{q_0}\[{(1+{\left(\frac{q_0{s}_0-F_3^{\′}}{H}\right)}^2)}^{1/2}-{(1+{\left(\frac{F_3^{\′}}{H}\right)}^2)}^{1/2}\]---\left(21\right)$

It is zero potential energy level with face z=0, obtains its unit gravitional force U_gFor:

$U_{gi}=Mg{\∫}_0^{L_0}z\left(s_0\right)d{s}_0---\left(22\right)$

Above-mentioned step 103, specifically includes following steps:

Step 401: assume that antenna total quality uniform equivalence average quality in each panel point is m, quadrangle list Unit's sum is n, moves according to truss and can be calculated the development rate of each panel point:

Wherein θ is antenna expanded angle,For angle between truss element, L₂For vertical bar of truss length.

Then truss kinetic energy is

$T_s=m\underset{i=2}{\overset{n}{\Σ}}({v_{ix}}^2+{v_{iy}}^2+{v_{iz}}^2)---\left(24\right)$

Step 402: set antenna fully deployed time antenna total potential energy as zero, then the gravitional force of antenna truss can represent For:

$E_v=-\underset{i=1}{\overset{n}{\Σ}}{\mathrm{mgL}}_2\sin \mathrm{\θ}=-{\mathrm{nmgL}}_{2}sin\mathrm{\θ}---\left(25\right)$

In formula, g is acceleration of gravity；

Step 403: for arbitrary connecting rod, the x-axis being floating coordinate system with the line of two-end-point A, B, utilizes the right hand fixed Then determine y-axis；Now the shape function of rod member can be described as:

$\mathrm{\Φ}=\left(\begin{array}{ccc}0& \mathrm{\ϵ}& 0\\ l_{AB}(\mathrm{\ϵ}-2{\mathrm{\ϵ}}^2+{\mathrm{\ϵ}}^3)& 0& l_{AB}({\mathrm{\ϵ}}^3-{\mathrm{\ϵ}}^2)\end{array}\right)---\left(26\right)$

In formula, l_ABFor beam element length, ε=x/l_AB；

Being described as of floating coordinate system lower link elastic deformation:

q_{ft i}=[q_{ft i1} q_{ft i2} q_{ft i3}]^T (27)

Q in formula_{f i1}And q_{f i3}It is respectively A point and the corner of B point, q_{f i2}For the x direction displacement in floating coordinate system of B point；

So elastic potential energy at the arbitrary rod member of t is:

$E_{pi}=\frac{1}{2}\[\frac{4EI}{l_{ab}}(q_{fi1}^2+q_{fi1}\·q_{fi3}+q_{fi3}^2)+\frac{{\mathrm{EA}}_s}{l_{ab}}{q}_{fi2}^2\]---\left(28\right)$

In formula, I is the moment of inertia of cross section, and E is elastic modelling quantity, A_sFor cross-sectional area；

The most whole flexible antennas hoop truss system in the elastic potential energy of t system is:

$E_p=\underset{i=1}{\overset{3n}{\Σ}}{E}_{pi}---\left(29\right)$

Step 404: above-mentioned kinetic energy, elastic potential energy, gravitional force are substituted into Lagrangian second equation, just can get truss Kinetic model.

Above-mentioned step 104, specifically includes following steps:

Step 501: to arbitrary quadrilateral units A_iB_iC_iD_i, due to its brace A_iC_iFor expansion link, centre is worn driving rope and is made Obtain antenna to launch, brace A_iC_iThe effect of holding capacity hardly, therefore ignored process；Thus at either connector, should have two hinges Chain retrains, with C_i(B_i+1As a example by)；

Definition subscript⁽ⁱ _, ⁿ⁾Represent that the node in i-th quadrilateral units is at the n-th rod member floating coordinate system x_i,ny_i,nz_iUnder Coordinate describe, hinge C_i(B_i+1) first constraint equation at place:

$R_{Bi}^{(i,2)}+A_{Bi}^{(i,2)}{u}_{Bi}^{(i,2)}=R_{Ci}^{(i,3)}+A_{Ci}^{(i,3)}{u}_{Ci}^{(i,3)}---\left(30\right)$

Due to

Wherein

$r_{Bi}^{(i,2)}={\left(\begin{array}{ccc}{L}_2& 0& 0\end{array}\right)}^T---\left(32\right)$

$u_{Bif}^{(i,2)}={\mathrm{\Φ}}_{(\mathrm{\ξ}=1)}{q}_{Bif}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{q}_{Bif1}\\ q_{Bif2}\\ q_{Bif3}\end{array}\right]=\left[\begin{array}{c}{q}_{Bif2}\\ 0\\ 0\end{array}\right]---\left(33\right)$

According to the description of hoop truss displacement, have:

WhereinFor the angle of adjacent two quadrilateral units, θ is antenna expanded angle；

So having:

In like manner can obtain C_i(B_i+1) second, place constraint equation:

Step 502: definition A₁B₁For fixing bar, fixing bar does not consider plastic deformation, hinge A₁The constraint equation at place is:

$R_{A1}^{(1,1)}+A_{A1}^{(1,1)}{u}_{A1}^{(1,1)}=0^{(1,1)}---\left(37\right)$

Due to

$u_{A1}^{(1,1)}={\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(38\right)$

I.e. have:

$R_{A1}^{(1,1)}={\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(39\right)$

Hinge B₁The constraint equation at place is:

$R_{A1}^{(1,1)}+A_{A1}^{(1,1)}{u}_{A1}^{(1,1)}=R_{B1}^{(1,2)}+A_{B1}^{(1,2)}{u}_{B1}^{(1,2)}---\left(40\right)$

Due to

$u_{A1}^{(1,1)}={\left(\begin{array}{cc}{L}_1=0& 0\end{array}\right)}^T,u_{B1}^{(1,1)}{\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(41\right)$

I.e. have:

$R_{A1}^{(1,1)}+\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{L}_1\\ 0\\ 0\end{array}\right]=R_{B1}^{(1,2)}---\left(42\right)$

Step 503: set up mode according to equation (35), (36), (39) and (42) constraint equation, available integrated antenna Moving atom group, and the Jacobian matrix of constraint is obtained by differential

Step 504: by rope net kinetic model, truss kinetic model and the Jacobian matrix of constraintSubstitute into equation (1) i.e. can get the kinetics equation of integrated antenna:

$\frac{d}{dt}\left(\frac{\∂T_E}{\∂\stackrel{\·}{q}}\right)-\frac{\∂T_E}{\∂q}+\frac{\∂(U_E+U_g)}{\∂q}+C_q^T\mathrm{\λ}=Q---\left(43\right)$

Wherein λ is Lagrange multiplier.

Beneficial effects of the present invention: the invention have the advantage that 1) can be accurate to the antenna expansion process containing cable net structure Analyze, obtain the dynamic situation of change of rope net form；2) expansion process rope net active force change curve can be accurately obtained, analyze The impact that antenna is launched by rope net tension force non-linear factor, provides basis for deployable antenna motor with Control System Design, keeps away Exempt from expansion process antenna and launch unstable or incomplete phenomena.

Below with reference to accompanying drawing, the present invention is described in further details.

Accompanying drawing explanation

Fig. 1 cable elements tensioning state；

Fig. 2 cable element slack form (in xoz plane)；

Fig. 3 slack line unit infinitesimal；

Fig. 4 elastic link describes schematic diagram；

Fig. 5 truss element coordinate system describes schematic diagram；

The main flow chart of the Fig. 6 antenna Deployment Dynamic Analysis method containing cable net structure；

Fig. 7 builds rope net kinetic model procedure chart；

Fig. 8 builds antenna truss kinetic model procedure chart；

Fig. 9 sets up constraint combined antenna integral power model process figure；

Figure 10 unfolded reticular antenna schematic diagram；

Figure 11 the inventive method is applied to the rope net active force variation diagram carrying out emulating on certain deployable rope net antenna structure.

Detailed description of the invention

See Fig. 5, the invention provides a kind of antenna Deployment Dynamic Analysis method containing cable net structure, including walking as follows Rapid:

Step 101: select material parameter D and geometric parameter S, Suo Wangtuo of netted deployable antenna truss element and rope net Flutter structure, the initial position P of rope net node.

Step 102: according to Lagrangian second equation, build the kinetic model of rope net.

$\frac{d}{dt}\left(\frac{\∂T_E}{\∂\stackrel{\·}{q}}\right)-\frac{\∂T_E}{\∂q}+\frac{\∂U_E+U_g}{\∂q}=Q---\left(1\right)$

In formula, q,For the generalized coordinates chosen and the generalized velocity of correspondence thereof, t is the time, T_EFor the kinetic energy of system, U_EFor The elastic potential energy of system, U_gFor the gravitional force of system, Q is the generalized force that nonconservative force is corresponding.

Seeing Fig. 6, this step 102 specifically includes following steps:

Step 201: be can get arbitrary cable elements i by rope net topology relation, its two node is j (position coordinates P_j=[x_j y_j z_j]^T) and k (position coordinates P_k=[x_k y_k z_k]^T).The generalized coordinates describing unit i is:

q_i=[x_jy_jz_jx_ky_kz_k]^T (2)

Corresponding generalized velocity is:

${\stackrel{\·}{q}}_i={\[{\stackrel{\·}{x}}_j{\stackrel{\·}{y}}_j{\stackrel{\·}{z}}_j{\stackrel{\·}{x}}_k{\stackrel{\·}{y}}_k{\stackrel{\·}{z}}_k\]}^T---\left(3\right)$

The generalized coordinates then describing whole rope net system is:

Q=[x_ny_nz_n]^T, n=1,2 ..., N (4)

The generalized velocity describing whole rope net system is:

$\stackrel{\·}{q}={\[{\stackrel{\·}{x}}_n{\stackrel{\·}{y}}_n{\stackrel{\·}{z}}_n\]}^T,n=1,2,\mathrm{...},N---\left(5\right)$

Wherein N is rope net node total number；T is matrix transposition symbol；

Step 202: by the uniform mass M of cable elements i_iIt is equivalent to two-end-point lumped mass have:

$M_j=M_k=\frac{1}{2}{M}_i---\left(6\right)$

If the element number being connected with any node k is c_i, the unit sum being connected is c_N, then the kinetic energy T at node k_k For:

$T_k=\frac{1}{4}({{\stackrel{\·}{x}}_k}^2+{{\stackrel{\·}{y}}_k}^2+{{\stackrel{\·}{z}}_k}^2)\underset{i=1}{\overset{c_N}{\Σ}}{M}_{ci}---\left(7\right)$

The then kinetic energy T of rope net system_cFor:

$T_c=\underset{k=1}{\overset{N}{\Σ}}{T}_k---\left(8\right)$

Step 203: by comparing the relation of cable elements chord length and former length, it is judged that whether cable elements is in tensioning state.If Cable elements chord length is more than former length, cable elements tensioning, forwards step 204 to；If cable elements chord length is less than or equal to former length, cable elements pine Relax, forward step 205 to；(Fig. 1 is cable elements tensioning state)

Step 204: cable elements chord length is more than former length, and cable elements tensioning, its mechanical stae meets Hooke theorem；Wherein jk ' For state before cable elements generation elastic deformation, jk is the tensioning state after catenary elements stress；Now, cable elements elastic potential energy U_EiCan be reduced to:

$U_{Ei}=\frac{EA}{2L_{0i}}{(L_{jk}-L_{0i})}^2---\left(9\right)$

${L_{jk}}^2={(x_j-x_k)}^2+{(y_j-y_k)}^2+{(z_j-z_k)}^2---\left(10\right)$

Wherein E is elastic modelling quantity, and A is cable elements sectional area；

Unit gravitional force U_giFor barycenter potential energy:

$U_{gi}=M_{i}g\frac{z_j+z_k}{2}---\left(11\right)$

In formula, g is acceleration of gravity.

Pass directly to step 206.

Step 205: cable elements chord length is less than or equal to former length, cable element slack, is derived by slack line list by unit The elastic potential energy of unit and gravitional force.

Seeing Fig. 7, this step 205 specifically includes following steps:

Step 301: lax catenary elements is by uniform load q₀Time, unit is sagging trend.In Fig. 2, dotted line is stretched wire Line does not considers form during elastic deformation, and solid line is form during catenary consideration elastic deformation, and some S is a some S₀Elasticity is occurred to become Position corresponding after shape.T is the tension force on cable elements at any point S, a length of s of point from starting point to S, corresponding former a length of s₀。 Whole machine balancing equation both horizontally and vertically all meets for any point S:

$T\frac{dx}{ds}=H---\left(14\right)$

$T\frac{dz}{ds}=q_0{s}_0-F_3^{\′}---\left(15\right)$

Step 302: meet geometric constraint satisfaction at any point S:

${\left(\frac{dx}{ds}\right)}^2+{\left(\frac{dz}{ds}\right)}^2=1---\left(16\right)$

Assume that material meets Hooke's law:

$T=AE(\frac{ds}{{\mathrm{ds}}_0}-1)---\left(17\right)$

Equation (16) will be substituted into after equation (14) and (15) summed square, can obtain the rope tensility on any point S point:

$T\left(s_0\right)={\[H^2+{(q_0{s}_0-F_3^{\′})}^2\]}^{1/2}---\left(18\right)$

Wherein H, F₃' by the fundamental equation of unit, obtained by numerical solution.Fig. 3 is slack line unit infinitesimal.

Step 303: elastic potential energy dU on any point S can be obtained by strain energy formulation_E:

${\mathrm{dU}}_E=\frac{T{\left(s_0\right)}^2{\mathrm{ds}}_0}{2EA}---\left(19\right)$

Thus the elastic potential energy of cable elements ab is:

$U_{Ei}={\∫}_0^{L_{0i}}{\mathrm{dU}}_E={\∫}_0^{L_{0i}}\frac{T{\left(s_0\right)}^2}{2EA}{\mathrm{ds}}_0---\left(20\right)$

Step 304: by conversion dz/ds=(dz/ds₀)(ds₀/ ds), formula (15) and formula (17) obtain catenary Z is to coordinate:

$z\left(s_0\right)=\frac{F_3^{\′}{s}_0}{AE}(\frac{q_0{s}_0}{2F_3^{\′}}-1)+\frac{H}{q_0}\[{(1+{\left(\frac{q_0{s}_0-F_3^{\′}}{H}\right)}^2)}^{1/2}-{(1+{\left(\frac{F_3^{\′}}{H}\right)}^2)}^{1/2}\]---\left(21\right)$

It is zero potential energy level with face z=0, its unit gravitional force U can be obtained_gFor:

$U_{gi}=Mg{\∫}_0^{L_0}z\left(s_0\right)d{s}_0---\left(22\right)$

Step 206: the elastic potential energy of rope net system is:

$U_E=\underset{i=1}{\overset{N_e}{\Σ}}{U}_{Ei}---\left(12\right)$

The gravitional force of rope net system is:

$U_g=\underset{i=1}{\overset{N_e}{\Σ}}{U}_{gi}---\left(13\right)$

Wherein, N_eFor cable elements sum.

Step 207: rope net generalized coordinates, kinetic energy, elastic potential energy, gravitional force are substituted into formula (1), obtains the dynamic of rope net Mechanical model.

Step 103: utilize Ruili-Ritz method to carry out discrete to truss element, the kinetic model of derivation flexible truss.

See Fig. 8, this step 103, specifically include following steps:

Step 401: assume that antenna total quality uniform equivalence average quality in each panel point is m, quadrangle list Unit's sum is n, moves according to truss and can be calculated the development rate of each panel point:

Wherein θ is antenna expanded angle,For angle between truss element, L₂For vertical bar of truss length.

Then truss kinetic energy is

$T_s=m\underset{i=2}{\overset{n}{\Σ}}({v_{ix}}^2+{v_{iy}}^2+{v_{iz}}^2)---\left(24\right)$

Step 402: set antenna fully deployed time antenna total potential energy as zero, then the gravitional force of antenna truss can represent For:

$E_v=-\underset{i=1}{\overset{n}{\Σ}}{\mathrm{mgL}}_2\sin \mathrm{\θ}=-{\mathrm{nmgL}}_{2}sin\mathrm{\θ}---\left(25\right)$

In formula, g is acceleration of gravity.

Step 403: see Fig. 4, for arbitrary connecting rod, the x-axis being floating coordinate system with the line of two-end-point A, B, profit Y-axis is determined by the right-hand rule.Now the shape function of rod member can be described as:

$\mathrm{\Φ}=\left(\begin{array}{ccc}0& \mathrm{\ϵ}& 0\\ l_{AB}(\mathrm{\ϵ}-2{\mathrm{\ϵ}}^2+{\mathrm{\ϵ}}^3)& 0& l_{AB}({\mathrm{\ϵ}}^3-{\mathrm{\ϵ}}^2)\end{array}\right)---\left(26\right)$

In formula, l_ABFor beam element length, ε=x/l_AB。

Being described as of floating coordinate system lower link elastic deformation:

q_{ft i}=[q_{ft i1} q_{ft i2} q_{ft i3}]^T (27)

Q in formula_{f i1}And q_{f i3}It is respectively A point and the corner of B point, q_{f i2}For the x direction displacement in floating coordinate system of B point.

So elastic potential energy at the arbitrary rod member of t is:

$E_{pi}=\frac{1}{2}\[\frac{4EI}{l_{ab}}(q_{fi1}^2+q_{fi1}\·q_{fi3}+q_{fi3}^2)+\frac{{\mathrm{EA}}_s}{l_{ab}}{q}_{fi2}^2\]---\left(28\right)$

In formula, I is the moment of inertia of cross section, and E is elastic modelling quantity, A_sFor cross-sectional area.

The most whole flexible antennas hoop truss system in the elastic potential energy of t system is:

$E_p=\underset{i=1}{\overset{3n}{\Σ}}{E}_{pi}---\left(29\right)$

Step 404: above-mentioned kinetic energy, elastic potential energy, gravitional force etc. are substituted into Lagrangian second equation, just can get purlin The kinetic model of frame.

Step 104: build the constraint equation of truss element and rope net, the kinetic model of rope net Yu truss is carried out group Close.

Seeing Fig. 9, this step 104 specifically includes following steps:

Step 501: to arbitrary quadrilateral units A_iB_iC_iD_i(without fixing bar), such as Fig. 5, due to its brace A_iC_iIt is flexible Bar, centre wear driving rope antenna is launched, brace A_iC_iThe effect of holding capacity hardly, therefore ignored process.Thus it is arbitrary Should there be two hinge restrainings joint, with C_i(B_i+1As a example by).

Definition subscript⁽ⁱ _, ⁿ⁾Represent that the node in i-th quadrilateral units is at the n-th rod member floating coordinate system x_i,ny_i,nz_iUnder Coordinate describe, hinge C_i(B_i+1) first constraint equation at place:

$R_{Bi}^{(i,2)}+A_{Bi}^{(i,2)}{u}_{Bi}^{(i,2)}=R_{Ci}^{(i,3)}+A_{Ci}^{(i,3)}{u}_{Ci}^{(i,3)}---\left(30\right)$

Due to

Wherein

$r_{Bi}^{(i,2)}={\left(\begin{array}{ccc}{L}_2& 0& 0\end{array}\right)}^T---\left(32\right)$

$u_{Bif}^{(i,2)}={\mathrm{\Φ}}_{(\mathrm{\ξ}=1)}{q}_{Bif}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{q}_{Bif1}\\ q_{Bif2}\\ q_{Bif3}\end{array}\right]=\left[\begin{array}{c}{q}_{Bif2}\\ 0\\ 0\end{array}\right]---\left(33\right)$

According to the description of hoop truss displacement, have:

WhereinFor the angle of adjacent two quadrilateral units, θ is antenna expanded angle.

So having:

In like manner can obtain C_i(B_i+1) second, place constraint equation:

Step 502: definition A₁B₁For fixing bar, fixing bar does not consider plastic deformation, hinge A₁The constraint equation at place is:

$R_{A1}^{(1,1)}+A_{A1}^{(1,1)}{u}_{A1}^{(1,1)}=0^{(1,1)}---\left(37\right)$

Due to

$u_{A1}^{(1,1)}={\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(38\right)$

I.e. have:

$R_{A1}^{(1,1)}={\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(39\right)$

Hinge B₁The constraint equation at place is:

$R_{A1}^{(1,1)}+A_{A1}^{(1,1)}{u}_{A1}^{(1,1)}=R_{B1}^{(1,2)}+A_{B1}^{(1,2)}{u}_{B1}^{(1,2)}---\left(40\right)$

Due to

$u_{A1}^{(1,1)}={\left(\begin{array}{cc}{L}_1=0& 0\end{array}\right)}^T,u_{B1}^{(1,1)}{\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(41\right)$

I.e. have:

$R_{A1}^{(1,1)}+\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{L}_1\\ 0\\ 0\end{array}\right]=R_{B1}^{(1,2)}---\left(42\right)$

Step 503: set up mode according to equation (35), (36), (39) and (42) constraint equation, available integrated antenna Moving atom group, and the Jacobian matrix of constraint is obtained by differential。

Step 504: by rope net kinetic model, truss kinetic model and the Jacobian matrix of constraintSubstitute into equation (1) i.e. can get the kinetics equation of integrated antenna:

$\frac{d}{dt}\left(\frac{\∂T_E}{\∂\stackrel{\·}{q}}\right)-\frac{\∂T_E}{\∂q}+\frac{\∂(U_E+U_g)}{\∂q}+C_q^T\mathrm{\λ}=Q---\left(43\right)$

Wherein λ is Lagrange multiplier.

Step 105: this model is solved based on Newmark method, i.e. can get the motion process of rope net form with The active force situation of change that truss is produced by rope net.

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

Look for shape method to be applied in certain rope net deployable antenna structure emulating, as shown in Figure 10 by the present invention.Its In netted deployable antenna hoop truss unit number N=6, truss element cross bar L₁=1m, montant L₂=0.6m, all Suo Dan Unit uses aramid fiber material, and elastic modelling quantity is E=2 × 10¹⁰Pa, cross-sectional area is A=π/4 × 10^-6m³, cable elements number is 211.

Figure 11 gives and analyzes the calculated cable net structure of method respectively to upper truss proximally and distally by the present invention Five to the Suo Li situation of change of joint

To sum up, the present invention is based on elastic catenary model of element, the kinetics equation of cable net structure of having derived, and derives The kinetic model of flexible truss Cable Structure, obtains antenna integral power model with cable net structure compositional modeling.This invention energy Enough antennas accurately obtained under flexible truss acts on jointly with non-thread sex cords net launch dynamic response, for engineering design and improvement Antenna launches the planning control of process etc. provides effectively support.Its committed step is the bullet from infinitesimal angle derivation catenary elements Property potential energy and gravitional force, obtain rope net kinetic model accurately.

The invention have the advantage that 1) antenna containing cable net structure can be launched process Accurate Analysis, obtain rope net form Dynamic situation of change；2) can be accurately obtained expansion process rope net active force change curve, analyze rope net tension force non-linear because of The impact that antenna is launched by element, provides basis for deployable antenna motor with Control System Design, it is to avoid launch process antenna exhibition Open instability or incomplete phenomena.

In present embodiment, the part of not narration in detail belongs to the known conventional means of the industry, chats the most one by one State.Exemplified as above is only the illustration to the present invention, is not intended that the restriction to protection scope of the present invention, every and basis Invent within same or analogous design belongs to protection scope of the present invention.

Claims (5)

1. the antenna Deployment Dynamic Analysis method containing cable net structure, is characterized in that: comprise the steps:

Step 101: select netted deployable antenna truss element and the material parameter of rope net, geometric parameter, rope net topology structure, The initial position P of rope net node；

Step 102: according to Lagrangian second equation, builds the kinetic model of rope net:

$\frac{d}{dt}\left(\frac{\∂T_E}{\∂\stackrel{\·}{q}}\right)-\frac{\∂T_E}{\∂q}+\frac{\∂U_E+U_g}{\∂q}=Q---\left(1\right)$

In formula, q,For the generalized coordinates chosen and the generalized velocity of correspondence thereof, t is the time, T_EFor the kinetic energy of system, U_EFor system Elastic potential energy, U_gFor the gravitional force of system, Q is the generalized force that nonconservative force is corresponding；

Step 103: utilize Ruili-Ritz method to carry out discrete to truss element, the kinetic model of derivation flexible truss；

Step 104: build the constraint equation of truss element and rope net, the kinetic model of rope net with truss is combined；

Step 105: solve this model based on Newmark method, i.e. can get motion process and the rope net of rope net form The active force situation of change that truss is produced.

A kind of antenna Deployment Dynamic Analysis method containing cable net structure, is characterized in that: described Step 102 specifically includes following steps:

Step 201: be can get arbitrary cable elements i by rope net topology relation, its two node is j and k, and the position of its interior joint j is sat It is designated as P_j=[x_j y_j z_j]^T, the position coordinates of node k is P_k=[x_k y_k z_k]^T；The generalized coordinates describing cable elements i is:

q_i=[x_j y_j z_j x_k y_k z_k]^T (2)

Corresponding generalized velocity is:

${\stackrel{\·}{q}}_i={\[{\stackrel{\·}{x}}_j{\stackrel{\·}{y}}_j{\stackrel{\·}{z}}_j{\stackrel{\·}{x}}_k{\stackrel{\·}{y}}_k{\stackrel{\·}{z}}_k\]}^T---\left(3\right)$

The generalized coordinates then describing whole rope net system is:

Q=[x_n y_n z_n]^T, n=1,2 ..., N (4)

The generalized velocity describing whole rope net system is:

$\stackrel{\·}{q}={\[{\stackrel{\·}{x}}_n{\stackrel{\·}{y}}_n{\stackrel{\·}{z}}_n\]}^T,n=1,2,...,N---\left(5\right)$

Wherein N is rope net node total number；T is matrix transposition symbol；

Step 202: by the uniform mass M of cable elements i_iIt is equivalent to two-end-point lumped mass have:

$M_j=M_k=\frac{1}{2}{M}_i---\left(6\right)$

If the element number being connected with any node k is c_i, the unit sum being connected is c_N, then the kinetic energy T at node k_kFor:

$T_k=\frac{1}{4}({{\stackrel{\·}{x}}_k}^2+{{\stackrel{\·}{y}}_k}^2+{{\stackrel{\·}{z}}_k}^2)\underset{i=1}{\overset{c_N}{\Σ}}{M}_{ci}---\left(7\right)$

The then kinetic energy T of rope net system_cFor:

$T_c=\underset{k=1}{\overset{N}{\Σ}}{T}_k---\left(8\right)$

Step 203: by comparing the relation of cable elements chord length and former length, it is judged that whether cable elements is in tensioning state；If Suo Dan Unit's chord length, more than former length, cable elements tensioning, forwards step 204 to；If cable elements chord length be less than or equal to former length, cable element slack, Forward step 205 to；

Step 204: cable elements chord length is more than former length, and cable elements tensioning, its mechanical stae meets Hooke theorem；Wherein jk ' is rope State before unit generation elastic deformation, jk is the tensioning state after catenary elements stress；Now, cable elements elastic potential energy U_EiCan It is reduced to:

$U_{Ei}=\frac{EA}{2L_{0i}}{(L_{jk}-L_{0i})}^2---\left(9\right)$

L_jk ²=(x_j-x_k)²+(y_j-y_k)²+(z_j-z_k)² (10)

Wherein E is elastic modelling quantity, and A is cable elements sectional area；

Unit gravitional force U_giFor barycenter potential energy:

$U_{gi}=M_{i}g\frac{z_j+z_k}{2}---\left(11\right)$

In formula, g is acceleration of gravity.

Pass directly to step 206；

Step 205: cable elements chord length is less than or equal to former length, cable element slack, is derived by slack line unit by unit Elastic potential energy and gravitional force；

Step 206: the elastic potential energy of rope net system is:

$U_E=\underset{i=1}{\overset{N_e}{\Σ}}{U}_{Ei}---\left(12\right)$

The gravitional force of rope net system is:

$U_g=\underset{i=1}{\overset{N_e}{\Σ}}{U}_{gi}---\left(13\right)$

Wherein, N_eFor cable elements sum；

Step 207: rope net generalized coordinates, kinetic energy, elastic potential energy, gravitional force are substituted into formula (1), obtains the dynamics of rope net Model.

A kind of antenna Deployment Dynamic Analysis method containing cable net structure, is characterized in that: described Step 205, specifically includes following steps:

Step 301: lax catenary elements is by uniform load q₀Time, unit is sagging trend；Point S is some S₀Elasticity is occurred to become Position corresponding after shape；T is the tension force on cable elements at any point S, a length of s of point from starting point to S, corresponding former a length of s₀； Whole machine balancing equation both horizontally and vertically all meets for any point S:

$T\frac{dx}{ds}=H---\left(14\right)$

$T\frac{dz}{ds}=q_0{s}_0-F_3^{\′}---\left(15\right)$

Step 302: meet geometrical constraint at any point S:

${\left(\frac{dx}{ds}\right)}^2+{\left(\frac{dz}{ds}\right)}^2=1---\left(16\right)$

Assume that material meets Hooke's law:

$T=AE(\frac{ds}{{\mathrm{ds}}_0}-1)---\left(17\right)$

Equation (16) will be substituted into after equation (14) and (15) summed square, can obtain the rope tensility on any point S point:

T(s₀)=[H²+(q₀s₀-F₃′)²]^1/2 (18)

Wherein H, F₃' by the fundamental equation of unit, obtained by numerical solution；

Step 303: elastic potential energy dU on any point S can be obtained by strain energy formulation_E:

${\mathrm{dU}}_E=\frac{T{\left(s_0\right)}^2{\mathrm{ds}}_0}{2EA}---\left(19\right)$

Thus the elastic potential energy of cable elements ab is:

$U_{Ei}={\∫}_0^{L_{0i}}{\mathrm{dU}}_E={\∫}_0^{L_{0i}}\frac{T{\left(s_0\right)}^2}{2EA}{\mathrm{ds}}_0---\left(20\right)$

Step 304: by conversion dz/ds=(dz/ds₀)(ds₀/ ds), by formula (15) and formula (17) obtain catenary z to Coordinate:

$z\left(s_0\right)=\frac{F_3^{\′}{s}_0}{AE}\left(\frac{q_0{s}_0}{2F_3^{\′}}-1\right)+\frac{H}{q_0}\[{\left(1+{\left(\frac{q_0{s}_0-F_3^{\′}}{H}\right)}^2\right)}^{1/2}-{\left(1+{\left(\frac{F_3^{\′}}{H}\right)}^2\right)}^{1/2}\]---\left(21\right)$

It is zero potential energy level with face z=0, obtains its unit gravitional force U_gFor:

$U_{gi}=Mg{\∫}_0^{L_0}z\left(s_0\right){\mathrm{ds}}_0---\left(22\right)$

A kind of antenna Deployment Dynamic Analysis method containing cable net structure, is characterized in that: described Step 103, specifically includes following steps:

Step 401: assuming that antenna total quality uniform equivalence average quality in each panel point is m, quadrilateral units is total Number is n, moves according to truss and can be calculated the development rate of each panel point:

Wherein θ is antenna expanded angle,For angle between truss element, L₂For vertical bar of truss length.

Then truss kinetic energy is

$T_s=m\underset{i=2}{\overset{n}{\Σ}}({v_{ix}}^2+{v_{iy}}^2+{v_{iz}}^2)---\left(24\right)$

Step 402: set antenna fully deployed time antenna total potential energy as zero, then the gravitional force of antenna truss is represented by:

$E_v=-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{mgL}}_{2}sin\mathrm{\θ}=-{\mathrm{nmgL}}_{2}sin\mathrm{\θ}---\left(25\right)$

In formula, g is acceleration of gravity；

Step 403: for arbitrary connecting rod, the x-axis being floating coordinate system with the line of two-end-point A, B, utilize the right-hand rule true Determine y-axis；Now the shape function of rod member can be described as:

$\mathrm{\Φ}=\left(\begin{array}{ccc}0& \mathrm{\ϵ}& 0\\ l_{AB}(\mathrm{\ϵ}-2{\mathrm{\ϵ}}^2+{\mathrm{\ϵ}}^3)& 0& l_{AB}({\mathrm{\ϵ}}^3-{\mathrm{\ϵ}}^2)\end{array}\right)---\left(26\right)$

In formula, l_ABFor beam element length, ε=x/l_AB；

Being described as of floating coordinate system lower link elastic deformation:

q_{ft i}=[q_{ft i1} q_{ft i2} q_{ft i3}]^T (27)

Q in formula_{f i1}And q_{f i3}It is respectively A point and the corner of B point, q_{f i2}For the x direction displacement in floating coordinate system of B point；

So elastic potential energy at the arbitrary rod member of t is:

$E_{pi}=\frac{1}{2}\[\frac{4EI}{l_{ab}}(q_{fi1}^2+q_{fi1}\·q_{fi3}+q_{fi3}^2)+\frac{{\mathrm{EA}}_s}{l_{ab}}{q}_{fi2}^2\]---\left(28\right)$

In formula, I is the moment of inertia of cross section, and E is elastic modelling quantity, A_sFor cross-sectional area；

The most whole flexible antennas hoop truss system in the elastic potential energy of t system is:

$E_p=\underset{i=1}{\overset{3n}{\Σ}}{E}_{pi}---\left(29\right)$

Step 404: above-mentioned kinetic energy, elastic potential energy, gravitional force are substituted into Lagrangian second equation, just can get the dynamic of truss Mechanical model.

A kind of antenna Deployment Dynamic Analysis method containing cable net structure, is characterized in that: described Step 104, specifically includes following steps:

Step 501: to arbitrary quadrilateral units A_iB_iC_iD_i, due to its brace A_iC_iFor expansion link, centre is worn driving rope and is made sky Line launches, brace A_iC_iThe effect of holding capacity hardly, therefore ignored process；Thus should have two hinges at either connector about Bundle, with C_i(B_i+1As a example by)；

(i n) represents that the node in i-th quadrilateral units is at the n-th rod member floating coordinate system x to definition subscript_i,ny_i,nz_iUnder Coordinate describes, hinge C_i(B_i+1) first constraint equation at place:

$R_{Bi}^{(i,2)}+A_{Bi}^{(i,2)}{u}_{Bi}^{(i,2)}=R_{Ci}^{(i,3)}+A_{Ci}^{(i,3)}{u}_{Ci}^{(i,3)}---\left(30\right)$

Due to

Wherein

$r_{Bi}^{(i,2)}={\left(\begin{array}{ccc}{L}_2& 0& 0\end{array}\right)}^T---\left(32\right)$

$u_{Bif}^{(i,2)}={\mathrm{\Φ}}_{(\mathrm{\ξ}=1)}{q}_{Bif}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{q}_{Bif1}\\ q_{Bif2}\\ q_{Bif3}\end{array}\right]=\left[\begin{array}{c}{q}_{Bif2}\\ 0\\ 0\end{array}\right]---\left(33\right)$

According to the description of hoop truss displacement, have:

WhereinFor the angle of adjacent two quadrilateral units, θ is antenna expanded angle；

So having:

In like manner can obtain C_i(B_i+1) second, place constraint equation:

Step 502: definition A₁B₁For fixing bar, fixing bar does not consider plastic deformation, hinge A₁The constraint equation at place is:

$R_{A1}^{(1,1)}+A_{A1}^{(1,1)}{u}_{A1}^{(1,1)}=0^{(1,1)}---\left(37\right)$

Due to

$u_{A1}^{(1,1)}={\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(38\right)$

I.e. have:

$R_{A1}^{(1,1)}={\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(39\right)$

Hinge B₁The constraint equation at place is:

$R_{A1}^{(1,1)}+A_{A1}^{(1,1)}{u}_{A1}^{(1,1)}=R_{B1}^{(1,2)}+A_{B1}^{(1,2)}{u}_{B1}^{(1,2)}---\left(40\right)$

Due to

$u_{A1}^{(1,1)}={\left(\begin{array}{cc}{L}_1=0& 0\end{array}\right)}^T,u_{B1}^{(1,1)}{\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T---\left(41\right)$

I.e. have:

$R_{A1}^{(1,1)}+\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{L}_1\\ 0\\ 0\end{array}\right]=R_{B1}^{(1,2)}---\left(42\right)$

Step 503: set up mode according to equation (35), (36), (39) and (42) constraint equation, the motion of available integrated antenna Constrained equations, and the Jacobian matrix of constraint is obtained by differential

Step 504: by rope net kinetic model, truss kinetic model and the Jacobian matrix of constraintSubstitute into equation (1) i.e. The kinetics equation of available integrated antenna:

$\frac{d}{dt}\left(\frac{\∂T_E}{\∂\stackrel{\·}{q}}\right)-\frac{\∂T_E}{\∂q}+\frac{\∂(U_E+U_g)}{\∂q}+C_q^T\mathrm{\λ}=Q---\left(43\right)$

Wherein λ is Lagrange multiplier.