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Dynamic qualification of large space structures by means of modal coupling techniques
Acta Astronautica Vol.7, pp. 1179-1190
PergamonPressLtd.,1980. Printedin GreatBritain
Dynamic qualification of large space structures by means of modal coupling techniquest A. B E R T R A M Institut fiir Aeroelastik, Deutsche Forschungs- und Versuchsanstalt fiir Luft- und Raumfahrt, Bunsenstrasse 10, D-3400 Gottingen, F.R.G. (Received 16 January 1980)
Abstract--In this paper some problems are described which are expected to arise during dynamic qualification of future large space structures. It is shown that the methods applied today, are no longer sufficient. As conclusion, the concept of a qualification procedure is proposed, which considers the phase of launching, as well as the phase of mission in orbit. Introduction IN ORDER to ensure safety and reliability of a spacecraft structure it has to be p r o v e d that the structure can withstand all expected loads during launch and during its life in orbit. The subjects of this procedure of dynamic qualification follow in brief s u m m a r y : • determination of the dynamic characteristics of the structure; • prediction of the response of the structure due to certain cases of excitation and loads; • p e r f o r m a n c e of qualification tests. This process is a c o m b i n e d procedure of assumptions, calculations, tests and evaluation of data. The tests take the k e y position; a m o n g others, they are n e c e s s a r y for the following two reasons: • to obtain realistic and reliable data about the dynamic properties; • to verify the calculations. It will be shown later that the coupling between structural dynamics and control is an important design consideration; therefore, an exact knowledge about the dynamic behaviour is an absolute necessity for a successful development of a large space structure. Large space structures Properties and environment
Erectable platforms, deployable antennas and large solar arrays are the first milestones on the w a y to large space structures (LSS); they are the fundamental units for the d e v e l o p m e n t of nearly all large structures which will be designed in the next 30 years. First demonstrations for the a s s e m b l y of large structures in the low Earth orbit are projected by N A S A within the next 5 years. T h e y plan to erect a l0 × 30 m platform and to deploy a large antenna structure of 50-200 m in tPaper presented at the XXXth Congress of the International Astronautical Federation, Munich, F.R.G., 17-22 September 1979. 1179
1180
A. Bertram
diameter; Fig. 1 illustrates one concept for such a large antenna. These demonstrations will precede the launch of the 25-kW Power M o d u l e I s h o w n in Fig. 2--with two rolled-out solar array blankets of about 9 x 32 m. Figure 3 shows a rough picture of the increase of the dimensions of space structures to be built within the following three decades. The solar power satellite will become a large planar structure, and will be constructed during the end of this century. The surface, which carries solar cells for electrical power generating, requires an area of about 100 km 2. The depth of the truss will be about 500 m. The first fundamental f r e q u e n c y of the truss equipped with solar cell blankets will be less than 0.005 Hz (Nansen and Diramio, 1978). Due to the extreme physical dimensions and the low structural stiffness resulting therefrom, a number of loads will be relevant to the structure in orbit: loads which had no meaning for the spacecraft we have built until now. Large amplitudes and low eigenfrequencies will create new problems for control designers. The usual way to separate the frequencies of the structure and the bandwidth of the control system will be no more applicable, as stiffening the structure will increase weight and reducing the control-system bandwidth means a reduction of the performance, i.e. of the pointing accuracy. However~ in that high demands are made on the control-system it will be necessary to control not only the attitude, but the shape as well. This shape control may be realized by
Fig. 1. Concept for a large deployable antenna (ref.: Marshall Space Flight Center, NASA).
Dynamic qualifications of large space structures
1181
Fig. 2. 25 kW Power module (ref.: Marshall Space Flight Center, NASA).
means of passive elements like springs and dampers or by means of active elements like actuators. In each case the dynamic behaviour of these non-linear elements and its effect on the structure has to be thoroughly investigated. This is also valid for non-linear structural elements like joints for instance. Due to their low fundamental frequencies, these structures are extremely prone to low-frequent excitation. Low-frequent vibrations will be excited by the thermal cycle, for instance. Another kind of vibration will be induced by control impulses when solar arrays or antennas are aligned or by impulses and shocks during the assembly. Assembly in a low orbit and transfer to a higher orbit by means of some kind of trucks will create a lot of dynamic problems for the structure and for the control-system. To summarize: the necessity of considering the flexibility of the structure for the attitude control on the one hand and for the control of the dynamic
1182
A. Bertram
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behaviour by means of passive or active control elements on the other hand requires an exact knowledge about the dynamic properties of the structures.
Dynamic qualification The future large space structure will be launched in a folded configuration, deployed and assembled with other units in a low Earth orbit and then dragged into a higher orbit. A separation must be made between launch and in-orbit configuration, whereby the loads during assembly and mission will be the predominant ones for the design. Dynamic qualification of the launch configuration will not be very different from the one used for current large satellites. However, above a certain weight and above certain dimensions, the sine vibration test on a shaker, i.e. single point excitation, will be no more applicable for design verification and for response testing. Tests on the structure in mission configuration with realistic boundary conditions will generally be impossible on Earth. An optimized test/analysis approach has to be developed, adapted to the specific problems of LSS and must take into consideration the phase of launching, as well as the phase of mission in orbit. A proposal for a concept, being elaborated in the DFVLR-Institute for Aeroelasticity, follows. Fundamental, a modal survey test (MST) seems to be the most successful test in determining realistic dynamic data, It will be practicable for the qualification of the launch configuration. By means of the test data, dynamic response and load analyses will be performed. For response testing, a multipoint-excitation test has to be developed. Multi-point excitation must be preferred to basic excitation even with multi-degrees of freedom, bearing in mind the interface of SHUTTLE payloads to the cargo bay. Concerning the mission configuration, modal survey tests can be performed on parts of the structure. These parts must be selected according to the practicability of exposure to tests on Earth.
Dynamic qualifications of large space structures
1183
The dynamic behaviour of the complete structure will be determined analytically by coupling procedures, proceeding from the data obtained by tests on elemental components. These components may be complete substructures or structural elements like beams, struts, parts of blankets or junction points, for instance. Much care must go into the investigation of the joints especially on their non-linear effects on the boundary conditions of the coupling. The procedure of coupling of elemental components to a complete structure is similar to the finite element technique with the advantage of experimentally obtained data. Figure 4 shows an example of an erectable platform consisting of eight equal bays. Th~s particular framework is assembled from 57 parts (inclusive 23 joints) of 7 different types. The substructure technique is extremely effective when the components are repeatedly used, as demonstrated in Fig. 5. It shows an elemental unit for huge structures. Proposed solutions for some selected problems M o d a l description o f d y n a m i c s
The mathematical description of a dynamic system by means of modal degrees of freedom implies the advantage of reducing the effort of calculation. With the general assumption of linearity and the presence of viscous damping, the well-known system of equations for a dynamic system with n discrete degrees of freedom (DOF) reads m i i + c u + ku = f,
(1)
where m, c and k are quadratic matrices (order n) of mass, damping and
Fig. 4. Assemblyof a platform structure (ref.: RockwellInternational).
1184
A. Bertram
Fig. 5. Elemental unit for large space structures (ref.: ERNO). stiffness of the s y s t e m and u and f are vectors of the independent D O F (nodal point displacements) and of the external forces; both time-dependent. The n u m b e r of D O F , which describes the structure of the satellite M A R O T S was 2200 in the first version of the finite element (FE) model, which was later c o n d e n s e d to a b o u t 300. V e c t o r u can be described by means of the series expansion of the m o d e shapes ~ : u = ~, ~sq~(t); s = 1,2 . . . . . m, (2) S
respectively u = ~q,
(3)
with q as the vector of generalized c o o r d i n a t e s . c o n t a i n i n g the factors of the series expansion and m as the n u m b e r of m o d e shapes taken into account. The columns of the m o d e shape matrix • can be determined in the modal s u r v e y test or by m e a n s of an eigenvalue analysis b a s e d on the finite element model.
Dynamic qualifications o[ large space structures
1185
Introducing the modal transformation of eqn (3) into eqn (1) and premultiplying by qbr leads to Mil + C/I + Kq = F,
(4)
with M, C and K as the matrices of generalized masses, generalized damping and generalized stiffness. The independent DOF now become the generalized coordinates q instead of the displacements u. The system of equations has been reduced by this from n to m. The convergence of eqn (3) depends on the selection of the modes taken into account and on the accuracy of the determined modal data, which can be obtained by FE calculations or by modal survey tests. Both methods deliver the normal frequencies, the mode shapes and the generalized masses. Whereby the damping, a necessary input for dynamic response analyses, can be determined additionally by an MST. The application of MST techniques on aerospace structures is described in detail by Niedbal (1979). Modal survey tests on the satellite MAROTS have been performed in the DFVLR-Institute for Aeroelasticity (Bertram et al., 1978) under ESTEC-contract. Fifty-two modes have been determined in the range from 13 to 100Hz. The condensed FE model had at least 300DOF. Another example of the effectiveness of modal techniques is shown by Garba et al. (1976) in the project, Viking Orbiter. The FE model had 32,000 DOF; the modal one, 250. Modal coupling procedures A number of modal coupling methods have been published during the last 20 years. A survey on the customary ones is given by Bertram et al. (1977). Breitbach (1976) presented two modal coupling procedures based on experimental modal data, which are currently being elaborated for application on spacecraft structures in the DFVLR-Institute for Aeroelasticity under ESTECcontracts (Bertram, 1979). The equations of motion of a structure assembled from the substructures A and B--neglecting external forces--reads:
M/i + C/! + Kq = 0,
(5)
with
M = I_
,A
IM J
l_
I c J~
(6)
Significant in applying coupling procedures is to consider the stiffness relations in the interface between substructures. The following two special cases of coupling conditions as given in Fig. 6 are described by Breitbach (1976): • two corresponding DOF are connected rigidly; • a flexibility with or without damping is effective between two connected DOF in case of elastic coupling.
1186
A. Bertram
£1emeot
I
1
ei91d£ooplio9
£1ostZCooplio9
Fig. 6. Rigid and elastic coupling conditions.
Generally, components are connected by mixed coupling conditions, whereby some DOF are rigidly coupled and some are elastically coupled. In the case of the elastic coupling by means of an elastic coupling element with the stiffness matrix k aB, the stiffness energy U in the spring yields: 1 U -: • uJkABu~.
(7)
The components of the vector ue are the physical displacements with elastic coupling conditions. Using the modal transformation U e =
~q,
(8)
and applying Lagrange's operation the additional stiffness AK A K = (~3eT k A l~ (~3e
!9)
M~I+ C q + ( K + AK)q = 0.
(10)
is introduced into eqn (5)
The case of rigid coupling is defined by a compatibility condition between corresponding DOF as shown in eqn (11) u, A - u , H -- 0,
(I1)
whereby ura and UrH contain the physical displacements with rigid coupling conditions. Using modal description yields
[®,A i -q,,"l
[qa] =0,
(12)
Dynamic qualifications of large space structures
1187
respectively • ,q = 0.
(13)
Compatibility implies a reduction of the number of DOF by the number of rigid coupled physical DOF. Vector q in eqn (5) containing the independent DOF decreases to ~. This reduction is caused by performing a transformation with T according to q =T~.
(14)
One procedure to produce T is described by Breitbach (1976). Matrix qbr of eqn (13) may be subdivided into a square matrix ~s and into a residual one ~.
(15) We obtain qs = - ~ s - ' ~ l
(16)
and finally q=
-
q = T~1.
(17)
Walton and Steeves (1969) showed that T can be found by means of any modal matrix of ~ , r ~ r corresponding to zero eigenvalues. Applying the transformation of eqn (14) to eqn (5) yields: M ~ I + Ci1+ K ~ I = O .
(18)
The matrices M, C and /( are symmetric, but no more diagonal. They are generated analogous to the following matrix transformation JQ1 = T r M T .
(19)
Especially in cases of statically indeterminate coupling, when the number of compatibility conditions is very large, the loss of degrees of freedom may become too high. The following describes a procedure for advantageously combining a great number of connecting points together with less reduction of DOF. Let the deformations u, be described by means of a cubic or quadratic polynomial under consideration of the special condition of a least squares approximation. For the deformations u~ in the interface yields after quadratic approximation Uri = Pl "4-P2Xri q- p3X 2" (20)
1188
A. Bertram
or respectively (21)
t])r=Xrp ,
where ~ is a mode shape vector containing the displacements u,~ of only the rigid coupled junction points. Matrix X~ is defined as
Xr =
1
x2
x~ ~
1
xi
Xi2
(22) _1
p is the vector of polynomial coefficients
P=
P2
•
(23)
P3 Vector p has to be determined for any mode • for both substructures by means of an approximation analysis. For this purpose different standard program routines are available. The complete mode shape matrix ~r is built in the following way Or = XrP. (24) The columns of P are the vectors p as defined in eqn (23). Equation (12) can now be written [qbAi__
B qA
or respectively [ p A l _ pB]
= O.
(26)
The completion of the procedure occurs as described in eqns (16)-(19). The number of constraints was reduced from the number of rigid coupled junction points to three by means of a quadratic approximation of the discrete displacements. With this procedure, a less reduction of DOF is obtained. Modal survey tests have been performed on several substructures of the satellite MAROTS (as shown in Fig. 7) in the DFVLR-Institute for Aeroelasticity (Bertram et al., 1978) under ESTEC-contract in order to apply the elaborated modal coupling procedures. The analyses have not been finished yet.
Dynamic qualifications of large space structures
1189
ontennos y s t e m ~ ~.~poylood
solaro[rays
~
module
servc'modul e e
Fig. 7. Substructures of satellite MAROTS.
Tests
.As seen above, tests are performed for two purposes • determination of the dynamic characteristics, • verification of response analyses. Modal survey tests on the complete structure in launch configuration will be possible. There are no restrictions concerning dimensions or weight. In case of a very complicated structure, a small number of substructure tests may be necessary. The MST technique is described by Niedbal (1979). He also presents a procedure for response tests by means of the modal survey test devices. This procedure provides a multi-point-excitation test, because this kind of proposed test is not limited with referring to size and weight of the structures, and the requirement for realistic excitation can be met. The dynamic qualification of large space structures in mission configuration can also be performed by means of MST when the procedure, proposed in this paper is applied. Small components with sufficient stiffness and strength for testing on Earth will be investigated dynamically and statically with special regard to non-linearities; the same is valid for joints and all kinds of connection elements. Test boundary conditions and test specification have yet to be developed. To verify the mathematical model, a simplified dynamic test should be performed in orbit, in order to exclude effects of preloads due to 1-g gravity and effects of covibrating air. Generally, it will be sufficient to test only a smaller representative part of the LSS. Two kinds of tests are imaginable. A MST procedure with the aim of determining some normal modes or--perhaps as a supplement--a response test, whereby the structure is exposed to any kind of excitation. The response of the structure, recorded in orbit will then be correlated with results of response analyses. Furthermore, normal mode parameters can be determined by means of these structural responses using improved modal survey test techniques. ~mmary
The concept, proposed in this paper, shows the possibility of qualifying even future large space structures by a procedure based on both, analyses and tests. This is an important advantage, as the damping properties of light structures cannot be determined by finite element calculations alone. However, it has to be
1190
A. Bertram
p r o v e d that the s u b s t r u c t u r e c o u p l i n g b y m e a n s of m e a s u r e d m o d a l d a t a will e n s u r e sufficient r e l i a b i l i t y e v e n in case of v e r y large a n d c o m p l i c a t e d s t r u c t u r e s .
References Bertram A., Degener M. and Freymann R. (1977) Development of modal techniques using experimental modal data. End of phase I report. D F V L R - A VA-Rep. I13 253-77 C 05. Bertram A., Hiiners H. and Niedbal N. (1978) Modal survey tests on MAROTS STM. Final Rep. D F V L R - A VA-Rep. IB 253-78 C 10. Bertram A. (1979) Development of modal techniques using experimental modal data. End of Phase I1 report. D F V L R - A VA-Rep. IB 253-79 C 09. Breitbach E. (1976) Investigation of spacecraft vibrations by means of the modal synthesis approach. ESA-SP 121, 1-7. Noordwiik/The Netherlands. Garba J. A., Wada B. K. and Chen J. C. (1976) Experiences in using modal synthesis within project requirements. Shock and Vibration Bulletin 46, 213-230. Nansen R. H. and Diramio H. (1978) Structures for solar power satellites. Astron. Aeron. 16(10), 55-59. Niedbal N. (1979) Obtaining normal mode parameters from modal survey tests. Proc. X X X t h Int. Astronautical Congress, Mfinchen. Walton W. C. Jr. and Steeves E. C. (1969) A new matrix theorem and its application for establishing independent coordinates for complex dynamical systems with constraints. N A S A TR R-326.
PergamonPressLtd.,1980. Printedin GreatBritain
Dynamic qualification of large space structures by means of modal coupling techniquest A. B E R T R A M Institut fiir Aeroelastik, Deutsche Forschungs- und Versuchsanstalt fiir Luft- und Raumfahrt, Bunsenstrasse 10, D-3400 Gottingen, F.R.G. (Received 16 January 1980)
Abstract--In this paper some problems are described which are expected to arise during dynamic qualification of future large space structures. It is shown that the methods applied today, are no longer sufficient. As conclusion, the concept of a qualification procedure is proposed, which considers the phase of launching, as well as the phase of mission in orbit. Introduction IN ORDER to ensure safety and reliability of a spacecraft structure it has to be p r o v e d that the structure can withstand all expected loads during launch and during its life in orbit. The subjects of this procedure of dynamic qualification follow in brief s u m m a r y : • determination of the dynamic characteristics of the structure; • prediction of the response of the structure due to certain cases of excitation and loads; • p e r f o r m a n c e of qualification tests. This process is a c o m b i n e d procedure of assumptions, calculations, tests and evaluation of data. The tests take the k e y position; a m o n g others, they are n e c e s s a r y for the following two reasons: • to obtain realistic and reliable data about the dynamic properties; • to verify the calculations. It will be shown later that the coupling between structural dynamics and control is an important design consideration; therefore, an exact knowledge about the dynamic behaviour is an absolute necessity for a successful development of a large space structure. Large space structures Properties and environment
Erectable platforms, deployable antennas and large solar arrays are the first milestones on the w a y to large space structures (LSS); they are the fundamental units for the d e v e l o p m e n t of nearly all large structures which will be designed in the next 30 years. First demonstrations for the a s s e m b l y of large structures in the low Earth orbit are projected by N A S A within the next 5 years. T h e y plan to erect a l0 × 30 m platform and to deploy a large antenna structure of 50-200 m in tPaper presented at the XXXth Congress of the International Astronautical Federation, Munich, F.R.G., 17-22 September 1979. 1179
1180
A. Bertram
diameter; Fig. 1 illustrates one concept for such a large antenna. These demonstrations will precede the launch of the 25-kW Power M o d u l e I s h o w n in Fig. 2--with two rolled-out solar array blankets of about 9 x 32 m. Figure 3 shows a rough picture of the increase of the dimensions of space structures to be built within the following three decades. The solar power satellite will become a large planar structure, and will be constructed during the end of this century. The surface, which carries solar cells for electrical power generating, requires an area of about 100 km 2. The depth of the truss will be about 500 m. The first fundamental f r e q u e n c y of the truss equipped with solar cell blankets will be less than 0.005 Hz (Nansen and Diramio, 1978). Due to the extreme physical dimensions and the low structural stiffness resulting therefrom, a number of loads will be relevant to the structure in orbit: loads which had no meaning for the spacecraft we have built until now. Large amplitudes and low eigenfrequencies will create new problems for control designers. The usual way to separate the frequencies of the structure and the bandwidth of the control system will be no more applicable, as stiffening the structure will increase weight and reducing the control-system bandwidth means a reduction of the performance, i.e. of the pointing accuracy. However~ in that high demands are made on the control-system it will be necessary to control not only the attitude, but the shape as well. This shape control may be realized by
Fig. 1. Concept for a large deployable antenna (ref.: Marshall Space Flight Center, NASA).
Dynamic qualifications of large space structures
1181
Fig. 2. 25 kW Power module (ref.: Marshall Space Flight Center, NASA).
means of passive elements like springs and dampers or by means of active elements like actuators. In each case the dynamic behaviour of these non-linear elements and its effect on the structure has to be thoroughly investigated. This is also valid for non-linear structural elements like joints for instance. Due to their low fundamental frequencies, these structures are extremely prone to low-frequent excitation. Low-frequent vibrations will be excited by the thermal cycle, for instance. Another kind of vibration will be induced by control impulses when solar arrays or antennas are aligned or by impulses and shocks during the assembly. Assembly in a low orbit and transfer to a higher orbit by means of some kind of trucks will create a lot of dynamic problems for the structure and for the control-system. To summarize: the necessity of considering the flexibility of the structure for the attitude control on the one hand and for the control of the dynamic
1182
A. Bertram
/0:
~
//71
J
S0~r
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//Nigh/
llluminotoc
10 O¸
1
/
10-1
oa
/
u~ 10~3¸
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5ETI
P°nS,tS'Vg~. s on .t,
~ RoSOurcos/PoUufioo ~
Observo#~
~0-/,.
/Appollo
~5
J
Yeor
~gTo 798o ~9'go zobo Fig. 3. Increase of the dimensionsof space structures.
196o
2o7o
behaviour by means of passive or active control elements on the other hand requires an exact knowledge about the dynamic properties of the structures.
Dynamic qualification The future large space structure will be launched in a folded configuration, deployed and assembled with other units in a low Earth orbit and then dragged into a higher orbit. A separation must be made between launch and in-orbit configuration, whereby the loads during assembly and mission will be the predominant ones for the design. Dynamic qualification of the launch configuration will not be very different from the one used for current large satellites. However, above a certain weight and above certain dimensions, the sine vibration test on a shaker, i.e. single point excitation, will be no more applicable for design verification and for response testing. Tests on the structure in mission configuration with realistic boundary conditions will generally be impossible on Earth. An optimized test/analysis approach has to be developed, adapted to the specific problems of LSS and must take into consideration the phase of launching, as well as the phase of mission in orbit. A proposal for a concept, being elaborated in the DFVLR-Institute for Aeroelasticity, follows. Fundamental, a modal survey test (MST) seems to be the most successful test in determining realistic dynamic data, It will be practicable for the qualification of the launch configuration. By means of the test data, dynamic response and load analyses will be performed. For response testing, a multipoint-excitation test has to be developed. Multi-point excitation must be preferred to basic excitation even with multi-degrees of freedom, bearing in mind the interface of SHUTTLE payloads to the cargo bay. Concerning the mission configuration, modal survey tests can be performed on parts of the structure. These parts must be selected according to the practicability of exposure to tests on Earth.
Dynamic qualifications of large space structures
1183
The dynamic behaviour of the complete structure will be determined analytically by coupling procedures, proceeding from the data obtained by tests on elemental components. These components may be complete substructures or structural elements like beams, struts, parts of blankets or junction points, for instance. Much care must go into the investigation of the joints especially on their non-linear effects on the boundary conditions of the coupling. The procedure of coupling of elemental components to a complete structure is similar to the finite element technique with the advantage of experimentally obtained data. Figure 4 shows an example of an erectable platform consisting of eight equal bays. Th~s particular framework is assembled from 57 parts (inclusive 23 joints) of 7 different types. The substructure technique is extremely effective when the components are repeatedly used, as demonstrated in Fig. 5. It shows an elemental unit for huge structures. Proposed solutions for some selected problems M o d a l description o f d y n a m i c s
The mathematical description of a dynamic system by means of modal degrees of freedom implies the advantage of reducing the effort of calculation. With the general assumption of linearity and the presence of viscous damping, the well-known system of equations for a dynamic system with n discrete degrees of freedom (DOF) reads m i i + c u + ku = f,
(1)
where m, c and k are quadratic matrices (order n) of mass, damping and
Fig. 4. Assemblyof a platform structure (ref.: RockwellInternational).
1184
A. Bertram
Fig. 5. Elemental unit for large space structures (ref.: ERNO). stiffness of the s y s t e m and u and f are vectors of the independent D O F (nodal point displacements) and of the external forces; both time-dependent. The n u m b e r of D O F , which describes the structure of the satellite M A R O T S was 2200 in the first version of the finite element (FE) model, which was later c o n d e n s e d to a b o u t 300. V e c t o r u can be described by means of the series expansion of the m o d e shapes ~ : u = ~, ~sq~(t); s = 1,2 . . . . . m, (2) S
respectively u = ~q,
(3)
with q as the vector of generalized c o o r d i n a t e s . c o n t a i n i n g the factors of the series expansion and m as the n u m b e r of m o d e shapes taken into account. The columns of the m o d e shape matrix • can be determined in the modal s u r v e y test or by m e a n s of an eigenvalue analysis b a s e d on the finite element model.
Dynamic qualifications o[ large space structures
1185
Introducing the modal transformation of eqn (3) into eqn (1) and premultiplying by qbr leads to Mil + C/I + Kq = F,
(4)
with M, C and K as the matrices of generalized masses, generalized damping and generalized stiffness. The independent DOF now become the generalized coordinates q instead of the displacements u. The system of equations has been reduced by this from n to m. The convergence of eqn (3) depends on the selection of the modes taken into account and on the accuracy of the determined modal data, which can be obtained by FE calculations or by modal survey tests. Both methods deliver the normal frequencies, the mode shapes and the generalized masses. Whereby the damping, a necessary input for dynamic response analyses, can be determined additionally by an MST. The application of MST techniques on aerospace structures is described in detail by Niedbal (1979). Modal survey tests on the satellite MAROTS have been performed in the DFVLR-Institute for Aeroelasticity (Bertram et al., 1978) under ESTEC-contract. Fifty-two modes have been determined in the range from 13 to 100Hz. The condensed FE model had at least 300DOF. Another example of the effectiveness of modal techniques is shown by Garba et al. (1976) in the project, Viking Orbiter. The FE model had 32,000 DOF; the modal one, 250. Modal coupling procedures A number of modal coupling methods have been published during the last 20 years. A survey on the customary ones is given by Bertram et al. (1977). Breitbach (1976) presented two modal coupling procedures based on experimental modal data, which are currently being elaborated for application on spacecraft structures in the DFVLR-Institute for Aeroelasticity under ESTECcontracts (Bertram, 1979). The equations of motion of a structure assembled from the substructures A and B--neglecting external forces--reads:
M/i + C/! + Kq = 0,
(5)
with
M = I_
,A
IM J
l_
I c J~
(6)
Significant in applying coupling procedures is to consider the stiffness relations in the interface between substructures. The following two special cases of coupling conditions as given in Fig. 6 are described by Breitbach (1976): • two corresponding DOF are connected rigidly; • a flexibility with or without damping is effective between two connected DOF in case of elastic coupling.
1186
A. Bertram
£1emeot
I
1
ei91d£ooplio9
£1ostZCooplio9
Fig. 6. Rigid and elastic coupling conditions.
Generally, components are connected by mixed coupling conditions, whereby some DOF are rigidly coupled and some are elastically coupled. In the case of the elastic coupling by means of an elastic coupling element with the stiffness matrix k aB, the stiffness energy U in the spring yields: 1 U -: • uJkABu~.
(7)
The components of the vector ue are the physical displacements with elastic coupling conditions. Using the modal transformation U e =
~q,
(8)
and applying Lagrange's operation the additional stiffness AK A K = (~3eT k A l~ (~3e
!9)
M~I+ C q + ( K + AK)q = 0.
(10)
is introduced into eqn (5)
The case of rigid coupling is defined by a compatibility condition between corresponding DOF as shown in eqn (11) u, A - u , H -- 0,
(I1)
whereby ura and UrH contain the physical displacements with rigid coupling conditions. Using modal description yields
[®,A i -q,,"l
[qa] =0,
(12)
Dynamic qualifications of large space structures
1187
respectively • ,q = 0.
(13)
Compatibility implies a reduction of the number of DOF by the number of rigid coupled physical DOF. Vector q in eqn (5) containing the independent DOF decreases to ~. This reduction is caused by performing a transformation with T according to q =T~.
(14)
One procedure to produce T is described by Breitbach (1976). Matrix qbr of eqn (13) may be subdivided into a square matrix ~s and into a residual one ~.
(15) We obtain qs = - ~ s - ' ~ l
(16)
and finally q=
-
q = T~1.
(17)
Walton and Steeves (1969) showed that T can be found by means of any modal matrix of ~ , r ~ r corresponding to zero eigenvalues. Applying the transformation of eqn (14) to eqn (5) yields: M ~ I + Ci1+ K ~ I = O .
(18)
The matrices M, C and /( are symmetric, but no more diagonal. They are generated analogous to the following matrix transformation JQ1 = T r M T .
(19)
Especially in cases of statically indeterminate coupling, when the number of compatibility conditions is very large, the loss of degrees of freedom may become too high. The following describes a procedure for advantageously combining a great number of connecting points together with less reduction of DOF. Let the deformations u, be described by means of a cubic or quadratic polynomial under consideration of the special condition of a least squares approximation. For the deformations u~ in the interface yields after quadratic approximation Uri = Pl "4-P2Xri q- p3X 2" (20)
1188
A. Bertram
or respectively (21)
t])r=Xrp ,
where ~ is a mode shape vector containing the displacements u,~ of only the rigid coupled junction points. Matrix X~ is defined as
Xr =
1
x2
x~ ~
1
xi
Xi2
(22) _1
p is the vector of polynomial coefficients
P=
P2
•
(23)
P3 Vector p has to be determined for any mode • for both substructures by means of an approximation analysis. For this purpose different standard program routines are available. The complete mode shape matrix ~r is built in the following way Or = XrP. (24) The columns of P are the vectors p as defined in eqn (23). Equation (12) can now be written [qbAi__
B qA
or respectively [ p A l _ pB]
= O.
(26)
The completion of the procedure occurs as described in eqns (16)-(19). The number of constraints was reduced from the number of rigid coupled junction points to three by means of a quadratic approximation of the discrete displacements. With this procedure, a less reduction of DOF is obtained. Modal survey tests have been performed on several substructures of the satellite MAROTS (as shown in Fig. 7) in the DFVLR-Institute for Aeroelasticity (Bertram et al., 1978) under ESTEC-contract in order to apply the elaborated modal coupling procedures. The analyses have not been finished yet.
Dynamic qualifications of large space structures
1189
ontennos y s t e m ~ ~.~poylood
solaro[rays
~
module
servc'modul e e
Fig. 7. Substructures of satellite MAROTS.
Tests
.As seen above, tests are performed for two purposes • determination of the dynamic characteristics, • verification of response analyses. Modal survey tests on the complete structure in launch configuration will be possible. There are no restrictions concerning dimensions or weight. In case of a very complicated structure, a small number of substructure tests may be necessary. The MST technique is described by Niedbal (1979). He also presents a procedure for response tests by means of the modal survey test devices. This procedure provides a multi-point-excitation test, because this kind of proposed test is not limited with referring to size and weight of the structures, and the requirement for realistic excitation can be met. The dynamic qualification of large space structures in mission configuration can also be performed by means of MST when the procedure, proposed in this paper is applied. Small components with sufficient stiffness and strength for testing on Earth will be investigated dynamically and statically with special regard to non-linearities; the same is valid for joints and all kinds of connection elements. Test boundary conditions and test specification have yet to be developed. To verify the mathematical model, a simplified dynamic test should be performed in orbit, in order to exclude effects of preloads due to 1-g gravity and effects of covibrating air. Generally, it will be sufficient to test only a smaller representative part of the LSS. Two kinds of tests are imaginable. A MST procedure with the aim of determining some normal modes or--perhaps as a supplement--a response test, whereby the structure is exposed to any kind of excitation. The response of the structure, recorded in orbit will then be correlated with results of response analyses. Furthermore, normal mode parameters can be determined by means of these structural responses using improved modal survey test techniques. ~mmary
The concept, proposed in this paper, shows the possibility of qualifying even future large space structures by a procedure based on both, analyses and tests. This is an important advantage, as the damping properties of light structures cannot be determined by finite element calculations alone. However, it has to be
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A. Bertram
p r o v e d that the s u b s t r u c t u r e c o u p l i n g b y m e a n s of m e a s u r e d m o d a l d a t a will e n s u r e sufficient r e l i a b i l i t y e v e n in case of v e r y large a n d c o m p l i c a t e d s t r u c t u r e s .
References Bertram A., Degener M. and Freymann R. (1977) Development of modal techniques using experimental modal data. End of phase I report. D F V L R - A VA-Rep. I13 253-77 C 05. Bertram A., Hiiners H. and Niedbal N. (1978) Modal survey tests on MAROTS STM. Final Rep. D F V L R - A VA-Rep. IB 253-78 C 10. Bertram A. (1979) Development of modal techniques using experimental modal data. End of Phase I1 report. D F V L R - A VA-Rep. IB 253-79 C 09. Breitbach E. (1976) Investigation of spacecraft vibrations by means of the modal synthesis approach. ESA-SP 121, 1-7. Noordwiik/The Netherlands. Garba J. A., Wada B. K. and Chen J. C. (1976) Experiences in using modal synthesis within project requirements. Shock and Vibration Bulletin 46, 213-230. Nansen R. H. and Diramio H. (1978) Structures for solar power satellites. Astron. Aeron. 16(10), 55-59. Niedbal N. (1979) Obtaining normal mode parameters from modal survey tests. Proc. X X X t h Int. Astronautical Congress, Mfinchen. Walton W. C. Jr. and Steeves E. C. (1969) A new matrix theorem and its application for establishing independent coordinates for complex dynamical systems with constraints. N A S A TR R-326.