AIM-91-1248-CP SIMULATION OF THE SPACE STATION STRUT-OUT CONDITION

Paul A. Blelloch and Nadine M. Mack Kelly S. Carney SDRC Engineering Services Division, Inc. NASA Lewis Research Center

San Diego, California Cleveland, Ohio

ABSTRACT

A method is presented for reanalyzing a truss structure when one of the truss elements (struts) has failed. The method uses a modal model of the nominal structure coupled with a residual flexibility term to predict the effect of the failed strut without resolving the finite element model. By implementing the method as part of the transient simulation, it is feasible to consider a large number of potential strut failures with a minimum amount of extra effort. Preliminary application of the method to the Space Station indicates excellent agreement with results based on modifying and resolving the finite element model.

INTRODUCTION

Calculating structural Ioads for the Space Station Freedom is a challenging process due to the number of different flight configurations to be studied. A further complication is added when the strut-out condition is considered. The Space Station is designed to be structurally fault tolerant to the extent that it will function satisfactorily when any one strut in the main truss fails. This is known as the strut- out scenario. In order to verify the structural integrity of the Space Station under the strut-out scenario, the analyst must calculate loads not only for every Space Station configuration, but also for every possible failed strut Since the main truss includes hundreds of struts that may possibly fail, it is not feasible to undertake a complete analysis in each case. The purpose of this paper is to present a methodology for recalculating Space Station loads for the strut-out condition with a minimal amount of extra effort.

Two possible approaches were considered and rejected. The first was to use the sensitivity of the modal properties of the structure to incremental changes in properties of strut members to predict changes in loads. This is the design sensitivity approach, and while it can be implemented using MSC/NASTRANts design sensitivity capability [I]. early results showed that the variations in loads due to a strut-out condition are too nonlinear to be accurately predicted using first order sensitivities. The second

approach was to use reanalysis techniques previously developed for application to aircraft structural components by McDonnell Aircraft Company [2]. We call this the NASTRAN reanalysis approach. This technique uses the mode shapes of the original problem to reduce the size of the modified problem, thereby accelerating the calculation of modified modal properties. We found that the mode shapes of the original problem did not sufficiently represent the dynamics of the modified problem, resulting in inaccurate load predictions. The addition of residual flexibility terms to improve the accuracy of the modified solution proved difficult using the techniques previously developed. A more fundamental problem with this approach is that while the CPU time required to calculate modified modal properties is reduced, the number of steps in the reanalysis effort is not significantly less than in the original analysis method.

A third approach, presented here, was finally selected. It is similar to the "NASTRAN reanalysis" approach, but applied directly during the simulation process, sidestepping the need to explicitly calculate modified modal properties. In this case, a negative spring element is added to the modal model to cancel the contribution of the strut element to be removed. This element is added during the simulation process using normal modes plus a residual flexibility term to represent the structure. If the residual flexibility term is not included, significant inaccuracies in the calculation of loads for the modified model arise due to modal insufficiency, but the addition of the residual flexibility term results in very accurate results, while using relatively few modes. This method was implemented in both Mac I1 MATLABTM and in CO-ST-IN, a program designed to study control- structure interaction problems associated with the Space Station [3].

This paper is organized as follows: First we develop the method mathematically and indicate certain assumptions that can be made to simplify the equations. Next we explain the implementation of the method. We then present an example based on the Space Station MB-6 configuration that demonstrates the method. Finally, we draw some conclusions.

Release A: "Copyright c 1991 by the American Insitute of Aeronautics and Astronautics, Inc. All rights reserved."

DEVELOPMENT

This method uses the modal equations of motion for the nominal structure (no failed struts), along with some residual flexibility information to calculate the modified equations of motion and the resulting modified loads. We do this by adding loads to the model which exactly cancel the effect of the strut element to be removed.

Consider the original equations of motion in standard modal form:

where:

q - z - R -

rl -

r2 -

fl - f2 -

XI -

x2 - @ l - - Yll - - Y12 - k - @z - - Y2l -

- Y22 -

modal degree of freedom for the original model diagonal matrix of modal damping ratios diagonal matrix of modal frequencies (rad=) mode shape coefficients at locations of external inputs mode shape coefficients for relative axial deflection at element to be removed external forces exciting the structure axial force applied at ends of element to be removed external outputs of the suucture (typically the loads we're calculating) axial deflection of the element to be removed modal contribution to the external outputs

residual static conmbution to the external outputs due to external forces

residual static contribution to the external outputs due to forces at the removed element axial stiffness of the removed element (AE/I) modal conmbution to the axial deflection at the removed element

residual static contribution to the axial deflection of the removed element due to external forces

residual static contribution to the axial deflection of the removed element due to forces at the removed element

- - - - The yii, %2, y21, and y22 terms represent contributions to the static deflection of the structure not represented by the retained modes, sometimes called static correction terms [4]. They are calculated by subtracting the modal contribution to the static response from the exact static response as follows:

where: %I - Static response at external outputs due to

unit loads at external inputs Y12 - Static response at external outputs due to

unit axial load at removed element '4'21 - Axial force in removed element due to unit

loads at external inputs % - Axial force in removed element due to unit

axial load at removed element

Since the procedure developed here depends on accurately predicting the axial deflection of the removed element, it is essential to include the static correction terms. Even if modes are retained up to 20 Hz for the Space Station, these modes represent only a small amount of the axial deflection of elements in the truss since axial modes of the truss occur at very high frequencies. Results calculated without using the residual flexibility terms,therefore, are very inaccurate.

As the element is compressed it applies a positive axial restoring force to the rest of the structure. As the element is extended it applies a negative axial restoring force to the rest of the structure. We wish to apply an equal and opposite force to the structure which cancels the effect of the element. To do this we choose f2 as follows:

Solve (3) for f2 as follows:

Now substitute back into Equation (1). to calculate the modified equations of motion:

For the specific application addressed in this paper, we are interested in calculating loads in the Space Station photovoltaic arrays. Since forces applied internally to the truss have no effect on loads in the - photovoltaic arrays, y,, = 0. Another simplification can be made by noting that the axial force in the removed element is equal to the axial stiffness of the element times the axial deflection. This implies that r2 = M. Making these two changes, the equations reduce to:

These equations are no longer uncoupled, since removing the element modifies the modal properties of the structure. They can either be simulated as they stand, or they can be diagonalized to calculate the modified modal properties before simulation. The modified modal properties are calculated by solving the following eigenvalue problem:

where: V - quare matrix of eigenvectors A - diagonal matrix of eigenvalues

To transform the modified equations of motion to modal coordinates, make the following substitution:

Assuming that the damping remains diagonal in the modified coordinates (a good assumption in the case of lightly damped modes), the modified equations of motion are:

- X l = *Vz + Yllfl (9b)

These are standard uncoupled modal equations of motion and can be simulated using the same methods used to simulate the equations of motion for the nominal system. The implementation of the method is discussed in the following section.

IMPLEMENTATION

This methodology was fust implemented and verified in Mac II MATLABTM (a general purpose interactive matrix manipulation program) and then implemented in CO-ST-IN . CO-ST-IN is a program designed to facilitate the analysis of control-structure interaction problems associated with the Space Station. It includes a simulation capability which is used to calculate loads in the Space Station PV arrays, both with and without active control systems. Mode acceleration data recovery is used to recover internal loads in the structure.

The standard method for simulating a transient response using CO-ST-IN is illustrated in Figure 1.

Modal data includes the matrices R and l-1, while the data recovery matrices are @ 1 and Y 1. In order to apply the method we also need the matrices 0 2 , Y21. and '3'22. These are generated in one additional static solution where the input loads include the external inputs (fl) as well as unit axial loads (f2) applied at each strut that may potentially fail, and outputs are axial element forces (x2) in each strut that may potentially fail. If the number of potential strut failures is large, a significant effort is involved in defining a unit axial load corresponding to each strut. In order to facilitate this, a preprocessor (ELMREM) was written. The input to ELMREM is a list of potential strut failures (element numbers). The output is a list of NASTRAN input loads corresponding to each potential strut failure and an index file that helps CO-ST-IN keep track of which load corresponds to which element. The output of the static solution from MSC/NASTRAN now includes the matrices 0 2 , Y21. and '3'22.

To perform a strut-out simulation, CO-ST-IN extracts the appropriate @2, Y21, and Y22.mauices from MSC/NASTRAN output, based on user input of the element number for the failed strut. CO-ST-IN - - then calculates the residual flexibility terms y l , , y2, - and YP based on a set of retained modes (selected by the user), the coupled stiffness and force distribution matrices of Equation (6) and solves the eigenvalue

problem for the modified frequencies, force distribution, and data recovery matrices of Equation (9). These uncoupled, modal equations of motion are simulated using a closed-form discretized solution that can include active control systems. The output time histories or minimum/maximum values are available in various formats for postprocessing.

In addition to the steps required for a standard simulation, this implementation requires the execution of the ELMREM program to set up potential strut failures, a single static solution in MSCMASTRAN to calculate the residual flexibility terms, and a simulation in CO-ST-IN to predict loads for each failed strut. Only one additional static solution is required, independent of the number of potential strut failures. Almost the entire numerical part of the method is implemented during the simulation, where it is transparent to the user. Initial experience indicates that the extra CPU time required to calculate the modified modal equations is relatively short. The major practical difficulty is in handling the large amount of output data that can be generated when considering a number of potential strut failures.

EXAMPLE

As an example of the methodology, consider the MB- 6 configuration of the Space Station illustrated in Figure 2. We are interested in calculating bending moments at the base of each of the four arrays during docking of the shuttle. For simplicity we will assume that the shuttle applies a unit impulse in all three DOF at the docking port identified in Figure 2. Previous results also showed that removing element 538 (also identified in Figure 2) had a significant effect on the bending moments during docking, so we will predict the bending moments for a model which does not include element 538 using results for a model that does include element 538.

We start with a model including only 56 modes below 1Hz. The maximum bending moments in both planes for all four arrays (eight outputs) are plotted in Figure 3. The first bar (black) illustrates the bending moments for the nominal model (no failed struts). The second bar (white) illustrates the bending moments for the exact reanalysis where element 538 was removed from the finite element model and new modal properties were calculated. The third bar (gray) illustrates the predicted maximum bending moments based on the method presented here. The predicted loads (gray) are very close to the exact loads (white), but not identical. This shows that while the method does a "goodn job of predicting the change in response due to the removal of element 538, errors are evident due to modal truncation.

Adding another 44 modes below 5 Hz generates the results illustrated in Figure 4. The black, white, and gray bars are interpreted in the same way. Note that in this case the predicted peak loads for the modified model are almost exactly equal to the true peak loads for this model. This shows that the method does a "very good" job of predicting the change in response due to the removal of element 538 when modes up to 5 Hz are included. Further tests showed that including modes beyond 5 Hz did not appreciably improve results, and in fact, after about 15 Hz results deteriorated. While the reason for this is unclear, it can be hypothesized that it is due to a numerical problem inherent in the large number of modes representing the structure.

CONCLUSIONS

In conclusion, we have presented a method for rapidly reanalyzing the Space Station under a strut-out scenario. The method uses the modes of the nominal structure plus the residual flexibility at the failed strut to predict a modified modal model. Inclusion of the residual flexibility term is essential to generate accurate results.

The procedure has been developed for the specific situation of the Space Station truss but should prove applicable to a number of other problems.

The method is implemented as part of the simulation in a manner which is relatively transparent to the user. It does not require modification of the finite element model, and the recalculation of modal properties is performed automatically at a relatively small incremental cost in computation time.

Preliminary results for an MB-6 Space Station model show excellent agreement with a complete reanalysis. The method will be applied to other Space Station configurations with a larger number of potential failed struts as time allows.

ACKNOWLEDGMENT

This work was supported by the Space Station office at the NASA Lewis Research Center. We would especially like to acknowledge the contribution of Dr. James McAleese and Dr. Francis Shaker of the Engineering Directorate at NASA Lewis.

REFERENCES

1. Version 65 MSCINASTRAN Applications Manual

2. Wang, B.P., Caldwell, S.P., Smith, C.M., "Improved Eigensolution Reanalysis Procedures in Structural Dynamics," MSC/NASTRAN User's Conference. 1990. Paper No. 46.

3. "CO-ST-IN, Control-STructure-INteraction, User's Manual, Version 1.50," SDRC Report No. 40922-4, October 3rd. 1989.

4. BIelIoch P.A., "Calculation of Structural Dynamic Forces and Stresses Using Mode Acceleration," AIAA J. of Guidance, Control and Dynamics, Vol. 12. No. 5, Sep.-Oct., 1989, pp. 760-762.

CO-ST-IN

Figure 1. Procedun for simulating response in CO-ST-IN.

Element

Location

Figure 2. Simplifkd model of the Space Station MB-6 configuration (YZ orientation of the PV arrays).

Bending Moment 80

(in-lbs) 60

40

20

0

Bending Moment (in-lbs)

1 2 3 4 5 6 7 8 Output Number

Figure 3. Comparison ot maximum loads for 1 Hz model.

1 2 3 4 5 6 7 8 Output Number

Figure 4. Comparison to maximum loads for 5 Hz model.