CN103761368B - A kind of cable mesh reflector antenna surface accuracy and method for designing while tension state - Google Patents

Abstract

The present invention relates to a kind of cable mesh reflector antenna surface accuracy and method for designing while tension state, is characterized in that：At least comprise the steps：1）An initial balanced reference configuration is determined first, and configuration during as k=0, corresponding rope net node location are designated as X⁽⁰⁾, the setting-out length of rope section is designated as 2）When determining kth time iteration, rope net needs the modal displacement 3 for producing）On the basis of kth time poised state, when determining kth time iteration rope section tension force with regard to rope length sensitive matrix node location with regard to rope length sensitive matrix 4）The long correction of rope of kth time is solved using Optimized model；5）Rope section setting-out length correction after kth time iteration is 6）Repeat the process, until meeting the given condition of convergence.The method of the present invention can be in the case where antenna surface accuracy be ensured so that the tension state of rope net is uniform.

Description

A kind of cable mesh reflector antenna surface accuracy and method for designing while tension state

Technical field

The present invention relates to a kind of cable mesh reflector antenna structure, particularly a kind of cable mesh reflector antenna surface accuracy with Method for designing while power state.

Background technology

The national defence such as antenna structure is widely used in communicating, navigates, positions, radar, radio astronomy and national economy it is many Field.To detect small-signal or increase transmission power, the bore for not requiring nothing more than antenna is very big, and requires the precision of reflecting surface It is very high.The bore of generally antenna needs to reach more than ten meters, even more greatly.And the electrical property for needed for obtaining, often require that antenna Reflector precision reach the 1/50 of its operation wavelength, this requirement is very harsh.For example, operating frequency is 3GHz（S-band） Antenna, weigh antenna reflective face precision root-mean-square error should be generally less than 2mm.

Cable mesh reflector antenna is one of major trend of large aperture antenna development, is also spaceborne Large deployable antenna A kind of important form, have the advantages that simple structure, it is lightweight, be easily achieved heavy caliber, draw in small volume.However, rope net is anti- The design for penetrating surface antenna is extremely difficult.Firstly, since the surface of cable mesh reflector is spliced by the triangle plane of rope net Form, there is the errors of principles.This requires that the surface accuracy of wire side node is higher, and expected mean square root error is typically less than 2/3.Secondly as rope net is flexible structure, all rope sections necessarily be in tensioning state and the tension state of rope net will be as far as possible Uniformly, more preferable mechanical property could so be obtained.

As the mechanical model of cable net structure is complicated, modeling error is big and the coupling between the displacement of rope net and tension state By force.Therefore, very high surface accuracy should be reached, realizes that uniform tension state is cable mesh reflector Antenna Design simultaneously again In key issue.

The content of the invention

It is an object of the invention to provide a kind of cable mesh reflector antenna surface accuracy and method for designing while tension state, So as in the case where antenna surface accuracy is ensured so that the tension state of rope net is uniform.

The object of the present invention is achieved like this, sets while a kind of cable mesh reflector antenna surface accuracy is with tension state Meter method, is characterized in that：At least comprise the steps：

1）An initial balanced reference configuration, configuration during as k=0, corresponding rope net node location are determined first It is designated as X⁽⁰⁾, the setting-out length of rope section is designated as

2）When determining kth time iteration, rope net needs the modal displacement for producing

3）On the basis of kth time poised state, sensitive matrix of the rope section tension force with regard to rope length when determining kth time iterationSensitive matrix of the node location with regard to rope length

4）The long correction of rope of kth time is solved using following Optimized model

Wherein,For the long correction of rope of kth, T^(k)For the tension force group of each rope section in rope net Into vector, T^(k)For the expected value of rope section tension force, X^(k)For the position vector of wire side node, γ is the minimum rope section for allowing Power；

5）Rope section setting-out length correction after kth time iteration is

6）Repeat the process, until meeting the given condition of convergence.

Described step 1) in the determination of initial equilibration reference configuration include：

a）The Preliminary design of pure cable net structure；

b）The determination of antenna structure initial equilibration reference configuration.

Described a）The Preliminary design of pure cable net structure includes：

To pure cable net structure, if rope section sum is n, free node sum is m, is respectively positioned on rope net node during Preliminary design In its nominal position, then the equilibrium equation of wire side node is

AT=0（1）

Wherein, T is that n × 1 that each rope section tension force is constituted is vectorial, matrix A ∈ R^3m×nFor the balancing matrix of rope net；Each rope net 3 × 1 piecemeal A in node q and rope section j homography A_qj；When rope section j is connected with node q, A_qjIt is to go out from the node Send out, along the unit column vector in rope section j direction；When rope section j is not connected with node q, A_qjFor 3 × 1 null vectors.

For cable mesh reflector, formula（1）It is to owe fixed, there is multigroup tension force mode, thus can determine that one group of satisfaction Given design requirement and the most uniform initial tension of tension state；Method therefor is as follows：

From matrix theory, formula is met（1）Tension force T can be expressed as

T=null (A) × α（2）

Kernels of the null (A) for matrix A in formula, α are corresponding coefficient vector；Can be determined using following Optimized model One as far as possible uniform rope net tension state：

Find α=[α₁,α₂,...]^T

s.t. T≥γ

In formula,It is the vector being made up of rope section tension force average, and same group Rope section has identical average, realizes even tension with this；γ is the tension force lower limit for allowing.

After the rope section tension force balanced on given position determines, corresponding rope section setting-out length, correspondence jth is just obtained Root rope section, corresponding setting-out length is

Wherein, L_jFor the length after rope section tension deformation, ε_j=T_j/(EA)_jFor the strain of rope section, wherein T_jFor rope section Power, (EA)_jFor the axial tension rigidity of rope section.

Described b）The determination of antenna structure initial equilibration reference configuration includes：

Using step a）The rope section tension force of middle determination solves the integrally-built poised state of antenna as initial tension, should State is the initial equilibration reference configuration of antenna.

Described step 3）Sensitive matrix of the rope section tension force with regard to rope length when determining kth time iterationNode location with regard to The sensitive matrix of rope lengthWithProcess be：

It is analyzed from given poised state solving sensitivity information using calculus of finite differences based on nonlinear finite element, if Now node i (i=1,2 ..., position m) is x_0i, rope section j (j=1,2 ..., tension force n) be T_0j；Every time individually to the Long increment Delta L of j root ropes section one rope of applying_0j, carry out the current location x that statics Analysis obtain new poised state lower node_j With rope section tension force T_j；So just sensitive matrix can be obtained by calculus of finite differencesWithJth row, that is, have

Take j=1 successively, 2 ..., n, complete sensitive matrix is obtained.

Described step 2）When determining kth time iteration, rope net needs the modal displacement for producingThe step of include：

1）Rectangular coordinate system Oxyz is set up in the preferable paraboloidal apex of design requirement, z-axis is along parabola focal axis side To then corresponding parabola equation is represented by：

x²+y²=4fz（7）

To ensure good tension state, it would be desirable to which parabola equation is modified to：

x²+y²=4f (z+h)（8）

Wherein, h is side-play amount of the parabola vertex along z-axis；

2）To determine the adjustment amount of each node, online for rope i-th（I=1, wherein 2 ..., c, c are wanted to there is shape The interstitial content asked）Individual node A, if its coordinate in current location is [x, y, z]^T, from the adjustable Suo Fangxiang in point A edges to amendment Parabola afterwards makees intersection, meets at point C.

Unit vector at point A along adjustable Suo Fangxiang is designated as：

P=[p_x p_y p_z]^T (9)

Make AC=d, when node A is located above amendment parabola for just, positioned at lower section when be negative, the coordinate of this sampling point C can To be expressed as

As point C is located on amendment parabola, therefore its coordinate meets parabola equation（8）, that is, have

(x+p_xd)²+(y+p_yd)²=4f (z+p_zd+h) （11）

Because h and d are in a small amount, to formula（11）Launch and ignore second order in a small amount, obtain

x²+2xp_xd+y²+2yp_yD=4fz+4fp_zd+4fh （12）

Can then obtain

So, point A is expressed as relative to the form for correcting paraboloidal deviation and its direction availability vector：

D=pd（14）

As can be seen that the deviation of node is only relevant with the node coordinate before adjustment and paraboloidal correction h of amendment；

3）It is c to be set in interstitial content on reflecting surface, if clocking requirement these nodes are respectively positioned on amendment parabola, note

D in formula_iFor the position deviation vector of i-th node.Now corresponding root-mean-square error is：

Main target during Antenna Design seeks to make above-mentioned root-mean-square error minimum, and this can be put down by making which Side is minimum and obtain, even if following formula takes minima

Due to formula（17）In contain variable h, therefore its condition for taking minima is

Thus parabola repairing amount h now and corresponding modal displacement Δ is just obtained_cOccurrence；

4）According to the definition of sensitive matrix, the relation between the displacement of wire side node and the long adjustment amount of rope is

So, when being adjusted to the position of wire side node by change adjustment Suo Changdu, attainable modal displacement It is R^3cIn a sub-spaces, can be expressed as

V in formula_iFor sensitive matrixI-th row, α_iFor corresponding coefficient；

And during iteration, need the displacement that node is produced to must be positioned in the subspace, it is this by displacement_cTo the subspace In projected, have

Displacement U_sIt is the modal displacement for needing to produce when being adjusted.

Advantages of the present invention is as follows：

1）While the cable mesh reflector antenna surface accuracy for being carried and tension state in method for designing, by the shape face of antenna Used as equality constraint, rope section tension force does not relax as inequality constraints precision, and using Tension Evenness as object function, so Hard requirement during Antenna Design can either be met, design effect can be optimized again.

2）Amendment parabola is being introduced it is determined that often walking when rope net in iteration needs the modal displacement for producing, can not affected On the premise of surface accuracy, the uniformity that rope section tension force does not relax with tension state is effectively ensured.

3）It is determined that often walking modal displacement when rope net in iteration needs the modal displacement for producing to sensitive matrix subspace Projected, the modal displacement for so obtaining can effectively ensure that the convergence of iterative process.

4）Sensitive matrix is introduced in Optimized model, the computational efficiency being remarkably improved when antenna cable mesh reflector is designed.

Description of the drawings

Fig. 1 is pure cable net structure schematic diagram；

Fig. 2 is node to the paraboloidal displacement explanatory diagram of amendment；

Fig. 3 is the FEM (finite element) model figure of embodiment of the present invention rope net parabola antenna；

Fig. 4 is the pure rope mesh portions in embodiment of the present invention antenna；

Fig. 5 is the iterative process of reflecting surface surface accuracy；

Fig. 6 is the iterative process of the minimax warp tension ratio of each several part wire side.

Specific embodiment

With reference to embodiment accompanying drawing, the invention will be further described：

A kind of cable mesh reflector antenna surface accuracy and method for designing while tension state, at least comprise the steps：

1）Reflecting surface surface accuracy is to be on the basis of previous balanced reference configuration with the Iterative Design of tension state Carry out.Accordingly, it would be desirable to determine an initial balanced reference configuration, configuration during as k=0, corresponding rope net section first Point position is designated as X⁽⁰⁾, the setting-out length of rope section is designated as

2）When determining kth time iteration, rope net needs the modal displacement for producing

3）On the basis of kth time poised state, sensitive matrix of the rope section tension force with regard to rope length when determining kth time iterationSensitive matrix of the node location with regard to rope length

4）The long correction of rope of kth time is solved using following Optimized model

Wherein,For the long correction of rope of kth, T^(k)For the tension force group of each rope section in rope net Into vector,For the expected value of rope section tension force, X^(k)For the position vector of wire side node, γ is the minimum rope section for allowing Power；

5）Rope section setting-out length correction after kth time iteration is

6）Repeat the process, until meeting the given condition of convergence.

The method includes following several committed steps：

Described step 1) in the determination of initial equilibration reference configuration include：

a）The Preliminary design and b of pure cable net structure）The determination of antenna structure initial equilibration reference configuration.

a）The Preliminary design of pure cable net structure

For pure cable net structure as shown in Figure 1, if rope section sum is n, free node number is m.Rope is made during Preliminary design Net node is respectively positioned in its nominal position, then the equilibrium equation of wire side node is

AT=0（1）

Wherein, T is that n × 1 that each rope section tension force is constituted is vectorial, matrix A ∈ R^3m×nFor the balancing matrix of rope net；Each rope net 3 × 1 piecemeal A in node q and rope section j homography A_qj；When rope section j is connected with node q, A_qjIt is to go out from the node Send out, along the unit column vector in rope section j direction；When rope section j is not connected with node q, A_qjFor 3 × 1 null vectors.

For cable mesh reflector, formula（1）It is to owe fixed, there is multigroup tension force mode, thus can determine that one group of satisfaction Given design requirement and the most uniform initial tension of tension state；Method therefor is as follows：

From matrix theory, formula is met（1）Tension force T can be expressed as

T=null (A) × α（2）

Kernels of the null (A) for matrix A in formula, α are corresponding coefficient vector；Can be determined using following Optimized model One as far as possible uniform rope net tension state：

Find α=[α₁,α₂,...]^T

s.t. T≥γ

In formula,It is the vector being made up of rope section tension force average, and same group Rope section has identical average, realizes even tension with this；γ is the tension force lower limit for allowing.

After the rope section tension force balanced on given position determines, corresponding rope section setting-out length, correspondence jth is just obtained Root rope section, corresponding setting-out length is

Wherein, L_jFor the length after rope section tension deformation, ε_j=T_j/(EA)_jFor the strain of rope section, wherein T_jFor rope section Power, (EA)_jFor the axial tension rigidity of rope section.

b）The determination of antenna structure initial equilibration reference configuration includes：

Cable mesh reflector antenna structure is made up of pure rope net and its sub-truss two parts, should be whole as one during modeling Body is analyzing.

Using step a）The rope section tension force of middle determination solves the integrally-built initial equilibration ginseng of antenna as initial tension Examine configuration.

Due to step a）In do not account for the deformation of sub-truss, when carrying out unitary analysiss, to deviate which preferable for rope mesh-shaped Position, the uniformity of rope net tension force can also be deteriorated.But due to the deformation very little of truss, it is ensured that the reference configuration after solution In, the deviation very little of rope mesh-shaped and its ideal position, and tension state is also comparatively uniform.This is followed by reflecting surface The basis of surface accuracy and tension state iterated revision.

Described step 3）Sensitive matrix of the rope section tension force with regard to rope length when determining kth time iterationNode location is closed The sensitive matrix of Yu SuochangWithFrom definition：

Here sensitive matrix is solved using calculus of finite differences, gives the long increment of little rope, investigate node location and tension force Situation of change；Have the tendency of to diminish in view of rope section tension force after truss deformation even lax, using backward-difference method, that is, given Long increment Delta L of rope_0i＜ 0；Accordingly even when there is partial relaxation in original state rope net, iteration also ensure that using backward difference After occur without relaxation, and forward difference is then without this advantage.

Sensitivity information is solved using finite element method, is analyzed from given poised state, if now node i （I=1,2 ..., m）Position be x_0i, section rope j（J=1,2 ..., n）Tension force be T_0j.Individually apply one to jth root rope section every time Long increment Delta L of individual rope_0j, carry out the current location x that statics Analysis are balanced state lower node_jWith rope section tension force T_j.So Just sensitive matrix can be obtained by calculus of finite differencesWithJth row, that is, have

Take j=1 successively, 2,3 ..., n, complete sensitive matrix is obtained.

Non linear finite element analysis is carried out using ANSYS softwares herein, typically for the cable net structure for there are m root rope sections, is needed M non linear finite element analysis to be carried out, expends the time very much.For this purpose, proposing to realize that multiple statics are asked in the following way Solution：

i）Model is stored as into " .DB " model file, analysis every time uses " resume " order Restoration model, saves weight The time of new modeling.

ii）Ask difference repeatedly need to solve, propose that the storage parameter using ANSYS, recovery parametric technique are solved so that ANSYS is once called just can all to solve the result for needing.Committed step is

Described step 2）When determining kth time iteration, rope net needs the modal displacement for producingDetermination include：

i）Rectangular coordinate system Oxyz is set up in the preferable paraboloidal apex of design requirement, z-axis is along parabola focal axis side To then corresponding parabola equation is represented by

x²+y²=4fz（7）

Tension state for control cord net simultaneously causes adjustment amount less, it is contemplated that the deviation of parabolic focus can be pacified in feed Compensate during dress, it would be desirable to which parabola equation is modified to

x²+y²=4f (z+h)（8）

Wherein, h is side-play amount of the parabola vertex along z-axis, and when the value is different, the adjustment amount of wire side node is also differed.By Can change in feed location, therefore can be by h as variable, after suitable h values should be selected to cause adjustment, wire side shape is located at and refers to On fixed parabola, while the tension state of rope net is also than more uniform.

ii）To determine the adjustment amount of each node, online for rope i-th（I=1, wherein 2 ..., c, c are wanted to there is shape The interstitial content asked）Individual node A, if its coordinate in current location is [x, y, z]^T, from the adjustable Suo Fangxiang in point A edges to amendment Parabola afterwards makees intersection, meets at point C, as shown in Figure 2.In general, when changing the length of adjustable rope, the corresponding net of the rope Face node mainly produces the displacement along rope length direction, and negligible perpendicular to the displacement of rope length direction.

Unit vector at point A along adjustable Suo Fangxiang is designated as

P=[p_x p_y p_z]^T（9）

Make AC=d, when node A is located above amendment parabola for just, positioned at lower section when be negative.The coordinate of this sampling point C can To be expressed as

As point C is located on amendment parabola, therefore its coordinate meets parabola equation（8）, that is, have

(x+p_xd)²+(y+p_yd)²=4f (z+p_zd+h) （11）

Because h and d are in a small amount, to formula（11）Launch and ignore second order in a small amount, obtain

x²+2xp_xd+y²+2yp_yD=4fz+4fp_zd+4fh （12）

Can then obtain

So, point A is expressed as relative to the form for correcting paraboloidal deviation and its direction availability vector

D=pd（14）

As can be seen that the deviation of node is only relevant with the node coordinate before adjustment and paraboloidal correction h of amendment.

iii）It is c to be set in interstitial content on reflecting surface, if clocking requirement these nodes are respectively positioned on amendment parabola, Note

D in formula_iFor the position deviation vector of i-th node.Now corresponding root-mean-square error is

The target of surface accuracy adjustment is just so that above-mentioned root-mean-square error is minimum.This can by making its square of minimum and Obtain, even if following formula takes minima

Due to formula（17）In contain variable h, therefore its condition for taking minima is

Thus parabola repairing amount h now and corresponding modal displacement Δ is just obtained_cOccurrence.

iv）According to the definition of sensitive matrix, the relation between the displacement of wire side node and the long adjustment amount of rope is

So, when being adjusted to the position of wire side node by change adjustment Suo Changdu, attainable modal displacement It is R^3cIn a sub-spaces, can be expressed as

V in formula_iFor sensitive matrixI-th row, α_iFor corresponding coefficient；

And during iteration, need the displacement that node is produced to must be positioned in the subspace, it is this by displacement_cTo the subspace In projected, have

Displacement U_sIt is the modal displacement for needing to produce when being adjusted.

Simulation example

By taking AstroMesh cable mesh reflector antennas as an example, the parameter of antenna is：Bore 10m, online wrinkly face is away from 6m, biasing Apart from 6m, away from 40m, antenna height 1.2m, main rope segments are 10 to off line wrinkly face, and upper and lower wire side cable elements sum is 288, are erected It is 85 to hauling rope sum.The cross section of rope for circle, a diameter of 1.4mm, truss cross section be hollow circular-tube, internal diameter 14mm, External diameter 15.2mm.Fig. 3 is the FEM (finite element) model of the cable mesh reflector antenna, and Fig. 4 is pure rope mesh portions therein, the material of each several part Material parameter is shown in Table 1.

The material parameter of 1 antenna structure each several part of table

When designing while carrying out surface accuracy with tension state using the method for the invention, the iteration mistake of surface accuracy Journey is as shown in figure 5, iterative process such as Fig. 6 institutes of the minimax warp tension ratio of corresponding upper wire side, lower wire side and vertical hauling rope Show.It can be seen that shape surface error is larger before optimization, its root-mean-square error is 5.22mm, and 0.07mm is then reduced to after optimization.Optimization The online wire side of provinculum has slack line, and the uniformity of tension force is poor, and without slack line after optimization, and uniformity improves, and particularly goes up The minimax warp tension ratio of wire side has reached 2.3, and this result is ideal.This explanation the method can improve shape face essence simultaneously The uniformity of degree and rope net tension force.

Table 2 lists the forward and backward shape surface error of optimization, structural stress, the change of rope net tension force, maximum distortion and natural frequency Situation.In tolerance zone, natural frequency is slightly improved structural stress, is passed through it has been observed that first five first order mode is entirety Vibration.

The leading indicator contrast of the optimization front-end geometry of table 2

Claims (4)

1. a kind of cable mesh reflector antenna surface accuracy and method for designing while tension state, is characterized in that：At least include as Lower step：

1) an initial balanced reference configuration is determined first, configuration during as k=0, corresponding rope net node location are designated as X⁽⁰⁾, the setting-out length of rope section is designated as

2) when determining kth time iteration, rope net needs the modal displacement for producing

3) on the basis of kth time poised state, sensitive matrix of the rope section tension force with regard to rope length when determining kth time iterationSection Sensitive matrix of the point position with regard to rope length

4) the rope long correction secondary to solve kth using following Optimized model

$\begin{array}{cc}find& {\mathrm{\ΔL}}_0^{\left(k\right)}={\[{\mathrm{\ΔL}}_{01},{\mathrm{\ΔL}}_{02},...\]}^T\end{array}$

$\begin{array}{cc}\min & f={(T^{\left(k\right)}+\frac{\∂T}{\∂L_0}{\mathrm{\ΔL}}_0^{\left(k\right)}-{\stackrel{\‾}{T}}^{\left(k\right)})}^T(T^{\left(k\right)}+\frac{\∂T}{\∂L_0}{\mathrm{\ΔL}}_0^{\left(k\right)}-{\stackrel{\‾}{T}}^{\left(k\right)})\end{array}$

$\begin{array}{cc}s.t.& X^{\left(k\right)}+\frac{\∂X}{\∂L_0}{\mathrm{\ΔL}}_0^{\left(k\right)}=U_s^{\left(k\right)}\end{array}$

$T^{\left(k\right)}+\frac{\∂T}{\∂L_0}{\mathrm{\ΔL}}_0^{\left(k\right)}\≥\mathrm{\γ}$

Wherein,For the long correction of rope of kth time, T^(k)For the tension force composition of each rope section in rope net Vector,For the expected value of rope section tension force, X^(k)For the position vector of wire side node, γ is the minimum rope section tension force for allowing；

5) the rope section setting-out length correction after kth time iteration is

$L_0^{(k+1)}=L_0^{\left(k\right)}+{\mathrm{\ΔL}}_0^{\left(k\right)};$

6) repeat step 2 to 5, until meeting the given condition of convergence.

2. method for designing while a kind of cable mesh reflector antenna surface accuracy according to claim 1 and tension state, It is characterized in that：Described step 1) in the determination of initial equilibration reference configuration include：

A) Preliminary design of pure cable net structure；

B) determination of antenna structure initial equilibration reference configuration.

3. method for designing while a kind of cable mesh reflector antenna surface accuracy according to claim 1 and tension state, It is characterized in that：Described step 3) sensitive matrix of the rope section tension force with regard to rope length when determining kth time iterationNode location is closed The sensitive matrix of Yu SuochangProcess be：

It is analyzed from given poised state, if now solving sensitivity information using calculus of finite differences based on nonlinear finite element The position of node i is x_0i, the tension force of rope section j is T_0j；Individually give jth root rope section one rope of applying long increment Delta L every time_0j, carry out Statics Analysis obtain the current location x of new poised state lower node_jWith rope section tension force T_j；So, take i=1 successively, 2 ..., M, just can obtain sensitive matrix by calculus of finite differencesWithJth row, that is, have

${\[\frac{\∂X}{\∂L_0}\]}_{ij}=\frac{x_j-x_{0i}}{{\mathrm{\ΔL}}_{0j}},{\[\frac{\∂T}{\∂L_0}\]}_{ij}=\frac{T_j-T_{0i}}{{\mathrm{\ΔL}}_{0j}}---\left(6\right)$

Take j=1 again successively, 2 ..., n are obtained complete sensitive matrix.

4. method for designing while a kind of cable mesh reflector antenna surface accuracy according to claim 1 and tension state, It is characterized in that：Described step 2) rope net needs the modal displacement for producing when determining kth time iterationThe step of include：

1) the preferable paraboloidal apex in design requirement sets up rectangular coordinate system Oxyz, z-axis along parabola focal axis direction, then Corresponding parabola equation is represented by：

x²+y²=4fz (7)

To ensure good tension state, it would be desirable to which parabola equation is modified to：

x²+y²=4f (z+h) (8)

Wherein, h is side-play amount of the parabola vertex along z-axis；

2) it is the adjustment amount that determines each node, i-th node A, the wherein i=1 online for rope, 2 ..., c, wherein c are to have The interstitial content of shape need, if its coordinate in current location is [x, y, z]^T, from the adjustable Suo Fangxiang in point A edges to after amendment Parabola make intersection, meet at point C, the unit vector at point A along adjustable Suo Fangxiang be designated as：

P=[p_x p_y p_z]^T (9)

Make AC=d, when node A is located above amendment parabola for just, positioned at lower section when be negative, the coordinate of this sampling point C can be with It is expressed as

$\left\{\begin{array}{c}{x}_0=x+p_{x}d\\ y_0=y+p_{y}d\\ z_0=z+p_{z}d\end{array}\right.---\left(10\right)$

As point C is located on amendment parabola, therefore its coordinate meets parabola equation (8), that is, have

(x+p_xd)²+(y+p_yd)²=4f (z+p_zd+h) (11)

Because h and d are in a small amount, launch and ignore second order to formula (11) in a small amount, obtain

x²+2xp_xd+y²+2yp_yD=4fz+4fp_zd+4fh (12)

Can then obtain

$d=-\frac{x^2+y^2-4fz}{2{\mathrm{xp}}_x+2{\mathrm{yp}}_y-4{\mathrm{fp}}_z}+\frac{4f}{2{\mathrm{xp}}_x+2{\mathrm{yp}}_y-4{\mathrm{fp}}_z}h---\left(13\right)$

So, point A is expressed as relative to the form for correcting paraboloidal deviation and its direction availability vector：

D=pd (14)

As can be seen that the deviation of node is only relevant with the node coordinate before adjustment and paraboloidal correction h of amendment；

3) it is c to be set in interstitial content on reflecting surface, if clocking requirement these nodes are respectively positioned on amendment parabola, note

${\mathrm{\Δ}}_c={\[d_1^T,d_2^T,...,d_c^T\]}^T---\left(15\right)$

D in formula_iFor the position deviation vector of i-th node；Now corresponding root-mean-square error is：

${\mathrm{\δ}}_c=\sqrt{{\mathrm{\Δ}}_c^T{\mathrm{\Δ}}_c/c}---\left(16\right)$

Main target during Antenna Design seeks to make above-mentioned root-mean-square error minimum, and this can be by making its square most It is little and obtain, even if following formula takes minima

${\mathrm{\δ}}_c^2=\frac{1}{c}{\mathrm{\Δ}}_c^T{\mathrm{\Δ}}_c=\frac{1}{c}\underset{i=1}{\overset{c}{\Σ}}{d}_i^T{d}_i---\left(17\right)$

Due to containing variable h in formula (17), therefore its condition for taking minima is

$\frac{\∂{\mathrm{\δ}}_c^2}{\∂h}=0---\left(18\right)$

Thus parabola repairing amount h now and corresponding modal displacement Δ is just obtained_cOccurrence；

4) definition according to sensitive matrix, the relation between the displacement of wire side node and the long adjustment amount of rope is

$\mathrm{\Δ}X=\frac{\∂X}{\∂L_0}{\mathrm{\ΔL}}_0---\left(19\right)$

So, when being adjusted to the position of wire side node by change adjustment Suo Changdu, attainable modal displacement is R^3c In a sub-spaces, can be expressed as

$\mathrm{\Γ}=span\left(\frac{\∂X}{\∂L_0}\right)=\{{\mathrm{\α}}_1{v}_1+{\mathrm{\α}}_1{v}_1+...+{\mathrm{\α}}_n{v}_n|{\mathrm{\α}}_i\∈R\}---\left(20\right)$

V in formula_iFor sensitive matrixI-th row, α_iFor corresponding coefficient；

And during iteration, need the displacement that node is produced to must be positioned in the subspace, it is this by displacement_cCarry out in the subspace Projection, has

$U_s=\frac{\∂X}{\∂L_0}{\left(\frac{\∂X^T}{\∂L_0}\frac{\∂X}{\∂L_0}\right)}^{-1}\frac{\∂X^T}{\∂L_0}{\mathrm{\Δ}}_c---\left(21\right)$

Displacement U_sIt is the modal displacement for needing to produce when being adjusted.