Passive shape control of space antennas with truss support structures

Compurers

CLSrrucrures

Vol. 45, No. 2, pp. 297-305,

1992

Printedin Great Britain.

0

PASSIVE SHAPE CONTROL OF SPACE ANTENNAS TRUSS SUPPORT STRUCTURES A. B. TRAKt and R. J.

0045-7949/92 $5.00 + 0.00 1992 PergamonPressLtd

WITH

MELOSH~

TDepartment

IDepartment

of Civil Engineering, University of Maryland, College Park, MD 20742, U.S.A. of Civil and Environmental Engineering, Duke University, Durham, NC 27706, U.S.A. (Received 7 August 1991)

Abstract-Passive control is achieved by modifying the nodal coordinates of the structure such that the modified structure deforms to the original (unmodified) geometry under the given loading. The modified geometry is obtained by changing the coordinates iteratively, by an amount equal and opposite to the deflections that occurred under the given loading. For this purpose, the structural and material properties and the boundary conditions of the truss, and the load data are assumed given. The technique is illustrated in four cases. In Cases I-III a determinate truss is utilized. In Case I, a modified structural geometry and in Case II, a modified surface geometry under single loading condition is sought and achieved. In Case III, a single modified surface geometry, which would satisfy two loading conditions simultaneously, was sought and an average modified geometry was found. In Case IV, an indeterminate truss was investigated

under a single loading condition. In all case studies, the convergence to the modified structure was both rapid and monotonic.

INTRODUCHON

Antenna performance has been based on a formula by Ruze [l] in which it is related to the root-meansquare (rms) value of the surface distortion error. In most of the antenna designs the rms value is restricted to one-hundreth of the beam wavelength to ensure antenna performance. There are two approaches to obtain and maintain an accurate surface; the passive approach and the active approach. Currently, both approaches are used together to achieve an accurate shape. In passive methods, a stiff-enough structure is designed so that the surface will undergo small distortions under environmental effects. There have been extensive studies by Hedgepeth [2-61, Bush et al. [7], Fager and Garriott [8], Fager [9], Card et al. [lo] on different structural concepts and behaviors. Hedgepeth [3] concludes that for large diameter structures the most accurate shapes are obtained by tetrahedral trusses. In active methods, the residual distortions are corrected by a control system consisting of sensors, actuators and decision-making control units. Various analytical approaches can be found in the studies of Bushnell [I 11, Haftka [12, 131, Haftka and Adelman [14-161 and Weeks[17]. In recent years there have been attempts to develop integrated design techniques by Adelman and Padula [ 181, Baier and Helwig [19], Padula et al. 1201,Steinbach and Winegar (211 deal with the sizing of elements and controlling shape at the same time. This study addresses the shape control of a truss in which nodal coordinates are selected to compensate for deformations and consequent surface inaccuracies. For this purpose, a tetrahedral truss is utilized. CAS

4512-G

The structural and material properties and the boundary conditions of the truss, and the loading conditions are assumed given. The technique is illustrated in four cases. In Cases I-III a determinate and in Case IV an indeterminate truss is considered. PROBLEM STATEMENT

We assume that a truss is given which has been designed as the support structure for an antenna. The geometry of this truss will be referred to as the original geometry. Mathematically, the problem can be stated as follows: Given the truss design parameters, i.e., the nodes and connectivity of the truss members, the location of nodes, the cross-sectional areas of the members, elasticity modulus of the material, and the displacement and loading boundary conditions. Find the modified coordinates of the nodes such that ci+ui=c,, cj = constant,

i = l-s

j = I-(n - s)

u = [K(c)1- ‘P,

(1)

(2) (3)

where ci and ui are the modified nodal coordinates and deflections, respectively; ci, is the original nodal coordinates; s is the total number of modified nodes; cj is the unmodified nodal coordinates; u, K and c are the nodal deflections, structural stiffness matrix and nodal coordinates of the modified structure, respectively; p is the applied load vector and n is the total number of nodes in the structure. The ui vector is a partition of u corresponding to node i.

297

A. 9. TRAKand R. J.

298

Equation (1) represents the condition that the nodes whose coordinates are modified must return to their coordinates in the original structure when the loads are applied. Equation (2) represents the condition that other nodes must keep their original locations in space. Equation (3) represents the load-deflection and the equilibrium equations of the modified structure. These equations are nonlinear, since the structural stiffness matrix is a nonlinear function of the coordinates. There are a total of 3s unknowns (ci, i = 1, . . , s) in 3s nonlinear equations to be solved, represented by eqns (l)-(3). There might be more than one solution to the problem defined by eqns (l)-(3). Here, a solution will be sought which involves the smallest changes to the original geometry while minimizing the surface distortion. SOLUTION

OF THE PROBLEM

The simple iteration method is selected for solving eqns (l)-(3). The method, also known as Picard iteration or functional iteration, is based on rewriting a given function F(x) in terms of an iterating function f(x) and the variable x as in eqn (4). The variable x can be obtained from F(x) in an explicit manner F(x) = x - f(x)

(4)

and the solution can be obtained from F(x) = x - f(x) = 0.

(5)

where k = 1,2,3,. . . is the iteration step number. The condition represented in eqn (1) can be rewritten as i = l-s,

F(c, ) = c, - f(q) = 0,

original structure by an amount equal and opposite to the nodal deflections caused by the loads. CONVERGENCE

CRITERION

The Euclidian norm is selected as the convergence criterion. The condition for terminating the iteration is defined as Il[‘k’- rck+ ‘)I1r Q t,

r(k) = {c\k’,c$k’,. . . ,@‘}F, r(k+I)={CIk+l),CIk+‘),.

..,c;k+l)}F

and t is a positive real number. If eqn (6) is rewritten for lck + ‘)and substituted in inequality (7), the convergence criterion can be stated as II

+ ack)- L)ll EG L

ack)= {up), IIF), . . . , uf)}‘,

6, = {%, c*o, . . . , c,}‘. The left-hand side of inequality (8) and t are defined as the performance error and the tolerance of performance, respectively. ANALYSES

The truss selected for study is the IO-node tetrahedral truss of Fig. 1. It is constructed by connecting three tetrahedrons. They are (l-3-7-4), (2-7-8-5) and (7-6-10-9). The apexes (4,5,9) form the designated surface of which the performance will be investigated. The performance surface (4-5-9) is parallel to the other surface (l-3-6-10-8-2) which lies in the X-Y plane.

f(ci) = cio- uir i = l-s. procedure

in eqn (5) can be

c(i’)= f(#$ - 1’) or &’ = c.IO_ R!k I - ‘)9 i = l-s.

(6)

Physically, the iteration expressed in eqn (6) can be interpreted as changing the nodal coordinates of the

(8)

where

where

Then, the iterative expressed as

(7)

where

ILLUSTRATIVE

Then, the iterative algorithm can be set up as X(t)= f(X(k- I’),

MELOSH

Fig. I. Ten-node tetrahedral truss.

299

Passive shape control of antennas Table 1. Applied load data Loading condition II

Loading condition I Load (kips) Joint Direction

-4.0

3.0

2.0

4 1

4 2

5 1

-2.0

1.0

1.0

-1.0

2.0

-3.0

-1.5

-1.0

1.0

-1.5

-1.0

5 3

9 1

9 2

9 3

4 1

4 3

5 1

5 2

5 3

9 1

9 3

The truss shown in Fig. 1 is indeterminate if all the members have nonzero cross-sectional areas. The determinacy identification analysis which is given by Trak [22] reveals that the members (2-7), (S-10) and (4-6) are redundant. The structure is modified by assigning zero values for the crosssectional areas of these members. The method is implanted in the computer program given in Trak [22]. The selected load data is presented in Table 1. The loading conditions are chosen such that they would produce distortions large enough to test the reliability of the iterative procedure. In the literature, antenna performance is measured by the rms value of the surface distortion error. Mathematically, the performance measure defined in inequality (8) is similar to the rms definition. Hence, a realistic value for the tolerance of performance is chosen from the chart in [9] for the structure under study. It is taken as t = 10S5 in.

Results of Case I

The iteration history is shown in Fig. 2. The results indicate that a solution is obtained in five cycles. The convergence is monotonic and rapid. The percent change between the original and modified member lengths of each bar is listed in Fig. 3. It is observed that 21 out of 24 member lengths are modified by less than 0.3%. Three members are changed between 0.3 and 0.6%. The effect of the geometry change on the bar forces is shown in Fig. 4. The magnitude of the change in each bar force is given as a percentage of the maximum bar force of the original structure. The bar force of any member is altered less than 1.4% from its original value.

For these cases eqn (1) can be expressed as

For this case, eqn (1) becomes

ci+ui=ci,,,

i = l-10.

Equation (2) has no significance in this case. The nodal directions constrained by the boundary conautomatically satisfied by the ditions are load-deflection and equilibrium equations [eqn (3)]. The original geometry is chosen as the initial point of the iteration. Hence C’O’ = CL0 ) i = l-10.

0

I (Table 1) is employed for this

Case IIa and Case Zlb: Surface joints are free

Case I: All the joints are free

ci + ui = c,,

Loading condition case study.

(9)

1

i= 1,2,3,

where i = 1,2,3 corresponds to the surface joints 4, 5 and 9, respectively. They are assumed to represent the antenna surface. In the iteration process, eqn (2) has to be incorporated in the iteration procedure to ensure that joints other than 4, 5 and 9 do not change their original locations. Hence, their errors are not included in performance error calculations.

3

2

Iteration Fig. 2. Iteration history of Case I.

4

5

300

A. B. TEAKand R. J. 5

0.6%

F ;

0.5%

MELOSH

c $

0.4%

_._._

1

3

5

7

9

11

13

Member

15

I?

19

21

23

25

27

Number

Fig. 3. Member length changes in Case I.

Equation (9) is again chosen as the initial point of the iteration process. Two different loading conditions are used in each case.

Case Zlb: Loading condition II

The loading condition is given in Table 1. All the other properties are the same as before.

Case Ila: Loading condition I

Rest&s

The loading condition is given in Table 1. All the structural and material properties are same as before.

The iteration history is shown in Fig. 8. The solution is obtained in five steps. The convergence is again monotonic and rapid. The member length changes are shown in Fig. 9. The unchanged members are not included in the figure. It is seen from the figure that the member length changes are higher in this case. The maximum change is little over 3.5%. The effect of the geometry change on the bar forces is shown in Fig. 10. The maximum change is less than 3% of the maximum bar force of the original structure.

Results

ofCase

Ila

The iteration history is shown in Fig. 5. It is seen that the solution is obtained in five steps. The convergence is monotonic and rapid. The member length changes are shown in Fig. 6. The unchanged members are not included in the figure. In this case study, thirteen members are modified. The maximum m~ifi~tion is less than 1.4%. The effect of the geometry change on the bar forces is shown in Fig. 7. It is seen that the maximum change is less than 0.9% of the maximum bar force of the original structure.

of Case IIb

Case III: Muitiple loading condition

Two independent loading conditions are considered. The loading conditions given in Table 1 are used.

= 1.4% !! IL L m

1.2%

J

1.0%

I -

0.6%

i? 8 0.6% t m 0.4% C g$ 0.2% 5 xz.

00% -.-.--

1

3

5

7

9

11

13

Member

15

17

19

21

23

Number

Fig. 4. Effect of the geometry change on bar forces in Case 1.

25

27

301

Passive shape control of antennas

1

2

3

4

5

Iteration

Fig. 5. Iteration history of Case IIa. 5

1.4%

P

3

1.2%

Z f

1.0%

0 - 0.0% z '; e 0.6%

f = 0.4% b g 0.2% 1

3

5

7

9

11

13

15

17

19

21

23

25

27

Member Number

Fig. 6. Member length changes in Case Ha.

The case where all the joints are permitted to change locations are not considered since the major concern in antenna design is the surface accuracy, not the accuracy of the whole structure. Therefore, only the case which involves the changing

~0,0%~““I”“‘t~““?~“““” 1

3

5

7

9

11

of the nodal coordinates of the surface nodes are studied. In Cases IIa and IIb two different loadings were used and two different modified structures were obtained.

13

15

17

19

21

23

Member Number

Fig. 7. Effect of the geometry change on bar forces in Case IIa.

25

27

A. B. TRAK and R. J. MELO~H

302

2

1

3

4

5

Iteration Fig. 8. Iteration history of Case IIb.

1

3

5

7

9

11

13

Member

15

17

19

21

23

25

27

Number

Fig. 9. Member length changes in Case IIb.

The two modified structures are the unique solutions for these cases. This can be proven by considering the load-deflection equations represented in eqn (3) which are reproduced here Ku=p,

(10)

where K is the structural stiffness matrix (3n x 3n), u is the nodal deflections (3n x l), p is the load vector (3n x 1) and n is the number of joints in the structure. If the stiffness matrix is positive definite, there is a unique solution to eqn (10) for the given load vector. This fact is illustrated in Cases IIIa and IIIb in

Member

Number

Fig. IO. Effect of the geometry change on bar forces in Case IIb.

303

Passive shape control of antennas Table 2. Initial performance errors for Cases IIIa-IIId Loading condition II

Loading condition I

Performance error (in)

Mod. Case IIb (Case IIIa)

Average mod. (Case IIIc)

Mod. Case IIa (Case IIIb)

Average mod. (Case IIId)

4.8850

2.4391

4.5379

2.2679

Table 3. Initial performance errors in the average modified and original structures Loading condition II

Loading condition I

Performance error (in)

Original strut. (Case IIa)

Average mod. (Case IIIC)

Original strut. (Case IIb)

Average mod. (Case IIId)

2.8581

2.439 1

5.0847

2.2679

Trak [22]. It is concluded that a single modified geometry cannot be achieved to satisfy both of the loading conditions simultaneously. Therefore, neither the modified geometry of Case IIa for loading condition I nor the modified geometry of Case IIb for loading condition II can be used as a common solution. However, a geometry can be found between the two modified geometries such that the performance errors given in Cases IIIa and IIIb would be decreased. This geometry will be called the average modified geometry. The average modified geometry is defined as c,, = (ch, x c,,)/2,

i = l-n,

(11)

where c, is the nodal coordinates of the average modified geometry; q,,,, is the nodal coordinates of the modified geometry in Case IIa; cimzis the nodal coordinates of the modified geometry in Case IIb and n is the total number of joints in the structure. This concept will be illustrated by Cases 111~and IIId. The structure with the average geometry defined in eqn (1 1), is loaded by loading conditions I in Case 111~ and by loading condition II in Case IIId. The initial performance errors in Cases IIIa-IIId are

presented in Table 2. As it is seen from Table 2 that the initial error is decreased by 50% in the average modified geometry. It is also interesting to compare the initial errors of the average modified and original structures under loading conditions I and II. The results are summarized in Table 3. The results show that the difference between the errors of the average modified and the original structure for loading condition I is small. However for loading condition II, the error difference is 50%. It is seen that there remains some performance error which cannot be eliminated by the passive approach. It should be noted that for determinate trusses, if the members are also the actuators the original surface can be recovered without disturbing the stress distribution in the structure. In determinate trusses, all the self-equilibrating member forces are contained in their respective members. Case IV: Indeterminate trusses

In this case study, the algorithm will be applied to a statically indeterminate truss which is subject to a single loading condition. Only the surface shape control of the indeterminate truss will be investigated.

106

G

5 105

5 g 104 5 103 ii 2 e 10 @ : 10’

$ t

100 10-l 0

II 1

2

3 ltoratlon

Fig. 11. Iteration history of Case IV.

4

5

304

A. B. TRAK and R. J. MELOSH

1

3

5

7

9

11

13

Member

15

17

19

21

23

25

27

Number

Fig. 12. Member length changes in Case IV.

The truss used in this case study is obtained by re-defining the cross-sectional areas. The truss members which are given zero values for their cross-sectional areas, are assigned nonzero values to make the truss indeterminate. Also, the magnitude of the areas of all the truss members are changed, so the initial performance error would have a similar ma~itude as the dete~inate structure under the same loading condition. All the other structural and material properties are kept the same as before. For this case, loading condition II is employed. After several attempts, it is found that 1S in* crosssections result in an initial performance of 5.16 in under this loading condition. For the same loading condition, an initial performance error of 5.08 in is observed in the statically determinate case. It is accepted that the two initial errors are close enough and hence, the indeterminate truss with 1.5 in* crosssectional areas can be used for the investigation and the results can be compared with the determinate case.

Results

of Case ZY

The iteration history is shown in Fig. 11. The convergence is rapid and monotonic. The solution is attained in five steps. The member length changes are presented in Fig. 12. The unmodified members are not included in the figure. There are 15 members whose lengths are changed. The maximum modification is 4% of the original length. The effect of the geometry and stiffness change on the bar forces are shown in Fig. 13. The maximum change is about 5.5% of the maximum bar force of the original structure. CONCLUSIONS

In Cases I and II, the desired modified geometry of a dete~inate truss was sought under a single loading condition. In Case I, all the nodal coordinates were permitted to change. A modified geometry which would deform to its original shape was obtained.

_ _._._ 1

3

5

7

9

11

13

Member

15

17

19

21

23

25

Number

Fig. 13. Effect of the geometry and stiffness change on bar forces in Case IV.

27

Passive shape control of antennas In Case II, only the surface nodal coordinates were

permitted to change. A modified surface geometry which would deform to its original shape was obtained. Two cases with two different loading conditions were investigated. In both cases, the required modified geometry was obtained. In Case III, a modified surface geometry for the determinate truss was sought such that both loading conditions were satisfied simultaneously. The investigation showed that such a geometry could not be found. (An active control system is needed to obtain the prescribed geometry.) However, an average modified geometry could be obtained such that the actuator inputs of the active control system would be decreased. The average modified geometry was found by averaging the nodal coordinates of the modified structures, and the modified structures were obtained independently for each loading. Furthermore, any active control system, which employs the members as the actuators, can be used without disturbing the stress distribution in the structure. The case studies for the determinate truss showed that the change in the geometry was so small that the bar forces were not affected significantly; hence, a design check was not necessary. However, it should be noted that any re-design of any member could be done individually since the truss was chosen to be statically determinate. In Case IV, the surface shape control of an indeterminate truss was investigated under a single loading condition. A modified surface geometry which would deform to its original shape was searched and obtained. However, the comparison of the cases revealed that, the additional nonlinearity introduced by the member stiffnesses in the indeterminate truss affected the results; hence, a design check may be required. It is concluded that an average modified structure can also be obtained for an indeterminate truss under multiple loading conditions. The study also showed that the simple iteration method yielded a rapid and monotonic convergence. The convergence rate was approximately one digit per cycle. This study demonstrates that deformed geometries with perfect electromagnetic performance are possible with the modification of the coordinates of a truss under a single loading condition.

REFERENCES

1. J. Ruze, Antenna

tolerance theory-a review. Proc. IEEE 54, 633-640 (1966). 2. J. M. Hedgepeth, Critical requirements for the design of large space structures. NASA CR-3484 (1981).

305

3. J. M. Hedgepeth, Accuracy potentials for large space antenna reflectors with passive structures. J. Spacecraft and Rockers 19, 211-217 (1982). 4. J. M. Hedgepeth, Influence of fabrication tolerances on the surface accuracy of large antenna structures. AIAA Jnl 20, 680-686 (1982). 5. J. M. Hedgepeth and i. R. Adams, Design concepts for

lame reflector antenna structures. NASA CR-3663 (1983). 6. j. M.. Hedgepeth, Support structures for large infrared telescooes. NASA CR-3800 (1984). 7. H. G.-Bush, M. M. Mikulas Jr and W. L. Heard Jr, Some design considerations for large space structures, AIAA Jnl 16, 352-359 (1978). 8. J. A. Fager and R. Garriott, Large-aperture expandable truss microwave antenna. IEEE Trans. Antennas and Propagation AP-17, 452-458 (1969). 9. J. A. Fager, Large space erectable antenna stiffness reauirements. J. Soacecraft and Rockets 17.86-92 (1980). 10. M. F. Card, W.‘L. Heard Jr and D. L. Akin, Construdtion and control of large space structures. NASA TM-87689 (1986). 11. D. Bushnell, Control of surface configuration by application of concentrated loads. AIAA Jnl 17, 71-77 (1979). 12. R. T. Haftka, Optimum placement of controls for static deformations of space structures. AIAA Jnl 22, 1293-1298 (1984). 13. R. T. Haftka, Simultaneous analysis and design. AIAA Jnl 23, 1099-I 103 (1985). 14. R. T. Haftka and H. M. Adelman, An analytical investigation of shape control of large space structures by applied temperatures. AIAA Jnl23,450-457 (1985). 15. R. T. Haftka and H. M. Adelman. Selection of actuator locations for static shape control’of large space structures by heuristic integer programming. Comput. Sfruct. 20, 575-582 (1985). 16. R. T. Haftka and H. M. Adelman, Effect of sensor and actuator errors on static shape control for large space structures. AIAA Jnl 25, 134-138 (1987). 17. C. J. Weeks, Static shape determination and control for large space structures: II. A large space antenna. Dynamic Systems, Measurement and Control 106, 267-272 (1984). 18. H. M. Adelman and S. L. Padula, Integrated therma1structuralelectromagnetic design optimization of large space antenna reflectors. NASA TM-87713 (1986). 19. H. J. Baier and G. Helwig, Integrated design and analysis approach for large precision structures, a collection of technical papers. AIAAIASMEIASCEIAHS 26th Structures, Structural Dynamics and Materials Conference, Part 1, pp. 713-719, Orlando, FL (1985). 20. S. L. Padula. H. M. Adelman and M. C. Bailev. Integrated structural electromagnetic optimization of large space antenna reflectors. A Collection of Technical Papers AIAA/ASMEIASCE/AHS 28th Structures, Structural Dynamics and Materials Conference, Part 1, pp. 508517, Monterey, CA (1987).

21. R. E. Steinbach and S. R. Winegar, Interdisciplinary design analysis of a precision spacecraft antenna. A Collection of Technical Papers, AIAA/ASME/ ASCEIAHS 26th Strucrures, Structural Dynamics and Materials Conference, Part 1, pp. 704-712, Orlando, FL (1985). 22. A. B. Trak, Passive shape control of space antennas with determinate truss support structures. Ph.D. dissertation, Duke University, Durham, NC (1989).