INTRODUCTION

Space flight hardware can be exposed todifferent loading conditions resulting frommanufacturing, transportation, and lunch and in-orbit operation. During the launch, the spacecraftwill be exposed to random loads of both mechanicaland acoustic nature. Mechanical random loads arethe base acceleration at the interface between the

Material Science Research India Vol. 6(1), 41-52 (2009)

Frequency response analysis of a small satellitestructure under random vibration acceleration

loads using Mode superposition method

HAMID EMAMI and FOAD FARHANI

Department of Mechanical Engineering, Iranian Research Organizationfor Science and Technology (I.R.O.S.T), P.O. Box: 15815-3538, Tehran (Iran).

(Received: January 12, 2009; Accepted: April 10, 2009)

ABSTRACT

Launch event imposes critical loading conditions on the space flight hardware. Space structuresare exposed to different environments and consequently the space vehicle experiences various loadingconditions. The random vibration load, which is inputted mechanically and acoustically from rocketengines, is one of the major loads which must be considered in payload flight hardware structures. Inthis study two different approaches for calculating the frequency response of a small satellite structureunder random vibration acceleration loads have been considered. These approaches are based onfrequency response analysis using modal effective modes and effective masses related to these modes,and finite element spectrum analysis. The first approach considers the overall behavior of the structure.On the other hand, the second approach reveals the response of the structure on each desired node.The results of the first approach are dependent on the number of selected modes, and effective massesrelated to the modes. The analyses for small satellites show that the application of superpositiontechnique can result in an appreciable reduction in the calculation time, and helps in finding points onthe structure, which show the behavior of the center of gravity during the tests. In addition, it may alsobe used as approximation for modeling of satellite behavior, treated as concentrated mass, during thequick analyses at the preliminary design, and for the satellite and launcher coupled analyses. Theanalyses show that the satellite behavior under the effect of local modes can be different. This increasesthe error in this approach, especially for larger structures, which have structure elements with greaterinertial effects and consequently higher local effective masses. This approach also predicts the equivalentstatic load based on root mean square (rms) calculation of power spectral density response. Thispaper also considers the influence of effective mode selection criteria on the rms value of the response.

Key words: Random vibration, modal effective mass, structural dynamics, space structures.

launch vehicle and the spacecraft. The random loadsare due to several sources such as the interactionbetween the launch vehicle structure and the engineexhaust noise combination. Additionally, turbulentboundary layers and tank sloshing of launcher willintroduce random loads.

The random vibration loads acting onpayload flight equipment result from the sources of

42 Emami & Farhani, Mat. Sci. Res. India., Vol. 6(1), 41-52 (2009)

random loads and structural response of thelauncher structure. These high frequencydisturbances result in both mechanical andacoustic borne excitation, and occur during thelaunch phase only (lift-off and ascent), while themain engines are operating. Random vibrationloads do not occur at regular intervals for adefined period of time; and as the name implies,they are random in time of application andduration. Vibro-acoustic data has been acquiredfrom static firings, flight measurements, as wellas acoustic tests and large-scale random vibrationtests in many space programs. The randomvibration design environments for space flighthardware must be presented, in the appropriatelauncher documents, for several mountinglocations1. This method requires the experimenterto have a detailed finite element model of thecomponent. The method can be used to take theflexibility of the supporting structure into account.A dynamic response analysis is performed usingNASTRAN software. This approach can be usedto determine random load factors for use insubsequent static analyses. In this study, thespectrum analysis based on finite elementmethod and the superpositions of effective modesare considered.

Application of superposition method isbased on the use of effective masses concept forsystems with one degree of freedom², and extensionof this method to systems with n degrees of freedomin a special normal modes system for calculation ofreaction forces as the result of loads due to randomvibration3.

Results show that the modesuperposition method is a cost effective approachto calculate the overall behavior of structure. Alsothis method is a quick method to evaluate theresponse of structure, and the overall parameterslike damping value can be corrected easily duringthe test stages. This study also concerns thecriteria of selecting effective modes to get betterresults. The results of response and rms (rootmean square) value of response are comparedwith the spectrum analysis based on finiteelement method. Because of the fact that thesuperposition method calculates the center ofgravity (COG) response of structure, this method

can be used to find appropriate measuring pointsfor performing environmental tests.

Spectrum response analysisRandom response analysis is treated as

a data reduction procedure, which is applied to theresults of a frequency response analysis. First, thefrequency response analysis is performed forsinusoidal loading conditions { }P , at a sequenceof frequencies iω . Normal data reductionprocedures are then applied to the output of thefrequency response analysis module, resulting in aset of output quantities )( iju ω , corresponding toan output j. The calculations of power spectraldensities and autocorrelation functions for the outputquantities are performed in the random analysismodule of NASTRAN software4.

Figure 1 is a simplified flow diagram forthe random analysis module. The inputs to themodule are the frequency responses )( ijH ω ofquantities to loading conditions { }aP atfrequencies iω , and the auto- and cross-spectraldensities of the loading conditions aS and baS .The response quantities jS may be displacements,velocities, accelerations, internal forces, or stresses.In this study the responses are presented byacceleration.

Application of these frequency responsetechniques to the analysis of random processesrequires that the system be linear and that theexcitation be stationary with respect to time. Thetheory includes a few important theorems which willbe reviewed [5].

An important quantity in random analysistheory is the autocorrelation function )(τjR of aphysical variable ju , and is defined by:

...(1)

It is to be noted that )0(jR is the timeaverage value of

2ju , which is an important quantity

in the analysis of structural failure. The one-sidedpower spectral density (PSD) )(ωjS of ju is definedby:

43Emami & Farhani, Mat. Sci. Res. India., Vol. 6(1), 41-52 (2009)

...(2)

It may be shown (using the theory ofFourier integrals) that the autocorrelation functionand the power spectral density are Fouriertransforms of one another. Thus, the autocorrelationfunction can be defined in terms of frequencyresponse functions as:

...(3)

From the mean-square theorem, the rmsvalue ju is given as:

...(4)

According to this correlation, the rms valuecan be obtained from the area under the spectraldensity curve. This quantity is a suitable criterionfor approximation of the destructive energy due tothe random vibrations in the system response.

The transfer function theorem states that,if )( ijH ω is the frequency response of any physicalvariable ju , due to an excitation source, )(ωQ ,which may be a point force, a loading condition orsome other form of excitation, i.e., if:

...(5)

where )(ωju and )(ωQ are the Fouriertransforms of ju and )(ωQ , then the powerspectral density of the response )(ωjS , is relatedto the power spectral density of the source )(ωaS ,by the relation:

...(6)

Equation (6) is an important result becauseit allows the statistical properties (e.g., theautocorrelation function) of the response of a

system to random excitation to be evaluatedthrough the technique of frequency response.

In our analyses, performed on basis of theabove discussion, the frequency response of thesatellite for a steady random vibration excitation,with unity spectral density in the whole range ofinterest in this study, is estimated. In this mannerthe response for each structure node can beobserved. In view of linear nature of Eq. (6), theresponse of each node can be determined throughthe multiplication of this response with the inputspectral density.

Superposition TechniqueFor a one-oscillator system with a

resonance if the frequency response function(FRF) representing the transmissibility of the baseacceleration to the COG Acceleration is given by[6]:

...(7)

A base accelerated structure (of mass M)can be represented exactly by oscillators in parallelwith accordingly adjusted mass stiffness anddamping. For each resonant frequency if thetransmissibility factor is calculated over a specifiedfrequency range (i.e f varies from 20Hz to2000Hz) with Eq. (7) split into its real and imaginaryparts:

...(8)

44 Emami & Farhani, Mat. Sci. Res. India., Vol. 6(1), 41-52 (2009)

With the effective mass im of eachresonant frequency if the frequency response isthen determined over the specified frequency rangewith the summation over all resonant frequencies

if :

...(9)

This means that real and imaginary partsare summed separately after which these resultsare Root Sum Squared.

The Power Spectral Density outputrepresenting the CoG accelerations is determinedover the specified frequency range using thefollowing relation:

...(10)

Model DescriptionIn this work, double-module cubical

satellite structures with elliptical doors on allsides of the upper and lower modules, havingaccess to the structure internal space, have beenstudied (see Fig. 2). The double-moduleconfiguration provides a suitable layout for thegravity gradient boom (GG boom), and the twobattery packs placed in the upper module.Placement of battery packs in the upper modulehelps in thermal control of the sensitive batteries.Vibrat ion sensi t ive components such aselectronic boards are packed in an electronicbox, in contact with the base plate, in the lowermodule. In this location, the electronic boardswithstand the minimum vibrations transmittedfrom the launcher. In addition, due to the accessdoors placed on the upper and lower module,the structure has the required flexibility foroptimum component lay-out and easy access tothe components in various stages of assembly,integration and test.

On basis of results from previous study[7], suitable distribution of effective masses in theelliptical access doors produces the best response

for reduction of dynamic loads resulting fromrandom vibration and shock phenomenon.

This structure is modeled using Patran pre/post analysis software and Nastran for FEM solver.The modeling consists of both structural and non-structural components, so the mass budget ofsatellite is applied along with the structural modeling.A summary of types of elements used in thismodeling is listed in Table 1. Beam elements areused for modeling of both stiffeners and boltedjoints, solid elements are implemented in thepocketing and thick regions, shell element are usedfor modeling of plate components, the concentratedmass and MPC’s are used for considering the pointmass and connections, respectively.

RESULTS AND DISCUSSIONS

In order to calculate the response, theconstrained modal analysis is applied. The interfaceat the supporting region is fixed by fixing six degreesof freedom through the connecting points. Themodal analysis results show that the first modeoccurs in the lateral direction. The first modes inthree directions are approximately 50 Hz. The modaleffective masses are extracted based on Guyan [8]reduction technique. Table 2 lists the effective modesin translational directions, which are calculated fromthe finite element analysis.

In order to compare the results of the satellitestructure response, obtained from the two methods

Table 1: Type and number of elementsused in the finite element model

Element Name Element NumberType

Beam element CBAR 1002

Solid element CHEXA 2970

Solid element CPENTA 58

Concentrated mass CONM2 27

Shell elements CQUAD4 20111

Shell elements CTRIA3 492

Multi-Point Constraint RBE2 117

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Table 2: Summery of effective modes

Mode Freq [Hz] % of Total % of Total % of TotalMass NX Mass NY Mass NZ

1 50.22 33.13% <1.00 <1.002 52.46 <1.00% 37.67 <1.00%3 55.32 <1.00% <1.00% 32.25%4 77.08 7.01% <1.00% <1.00%5 92.11 <1.00% 3.52% <1.00%11 146.68 <1.00% <1.00% 15.46%16 160.72 <1.00% <1.00% 1.02%19 170.05 10.02% <1.00% <1.00%20 171.91 6.65% <1.00% <1.00%22 178.96 3.03% <1.00% <1.00%26 191.90 1.41% 1.08% <1.00%47 255.93 <1.00% 11.41% <1.00%49 261.64 <1.00% 1.72% <1.00%52 266.53 <1.00% 2.50% <1.00%60 301.29 1.62% <1.00% <1.00%61 305.45 <1.00% <1.00% 21.50%62 311.17 <1.00% <1.00% 1.25%69 319.01 <1.00% <1.00% 2.43%72 326.63 <1.00% <1.00% 1.90%96 409.37 <1.00% 1.24% <1.00%127 506.02 <1.00% <1.00% 5.33%202 710.02 1.32% <1.00% <1.00%214 749.40 <1.00% 2.36% <1.00%269 905.06 <1.00% 2.00% <1.00%270 906.41 <1.00% <1.00% <1.00%302 996.64 <1.00% 2.80% <1.00%303 997.60 <1.00% 4.44% <1.00%304 1000.43 <1.00% 1.56% <1.00%312 1016.70 8.34% <1.00% <1.00%403 1235.71 1.38% <1.00% <1.00%

considered in this study, input data on randomvibrations from Cosmos launcher InterfaceCompatibility Document (ICD) [9] has been used. Theproposed value and spectrum for this load case ispresented for launcher as given in Table 3. The rmsvalue (area under PSD curve) of the excitation is 7.97g.

The superposition technique has beenused to evaluate the frequency response functionof the satellite structure due to launcher input. Thecriteria for selection of effective masses are variedfrom 1 to 8% of whole structures.

Results (Figs. 3 through 5 (show that theamplitude of most effective mode shapes does notundergo considerable change. The mostconservative criteria for the effective mass areselected as 5%. One of the objectives of applyingthis method is to calculate the equivalent staticloads. The rms value of these curves can beconsidered as the equivalent static load that thestructure must undergo. It is seen that the modesin the lateral directions are more distributed thanthe axial modes.

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Fig. 2: The satellite structuremodel with elliptical access doors

Fig. 1: A simplified flow diagramfor the random analysis

Fig. 3: Response of the CoG (using the superposition method in X-direction)

Fig. 4: Response of the CoG (using the superposition method Z-direction)

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Fig. 5: Response of the CoG (using the superposition method in Y-direction)

Fig. 6: Response in X-direction due to excitation in X-direction at middle solar panels joint

Fig. 8: Response in Z-direction due to excitation in Z-direction at middle solar panels joint

Fig. 7: Response in Y-direction due to excitation in Y-direction at middle solar panels joint

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Fig. 9: Response in X-direction due to excitation in X-direction at the boom joint

Fig. 10: Response in Y-direction due to excitation in Y-direction at the boom joint

Fig. 11: Response in Z-direction due to excitation in Z-direction at the boom joint

Fig. 12: Response in X-direction due to excitation in X-direction at upper module stiffeners

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Fig. 13: Response in Y-direction due to excitation in Y-direction at upper module stiffeners

Fig. 14: Response in Z-direction due to excitation in Z-direction at upper module stiffeners

Figures 6 through 14 present response ofsatellite structure under random loading, at threemeasuring points in finite element model. One ofthe points is on the lateral side of the satellite onthe solar panels, the other is placed on thegravitational gradient boom, and the last is on theupper module stiffener. These results are comparedwith the results of superposition technique. Table 4presents the results at different points and centerof gravity.

Considering Patran software results andthe shape of main structure modes reveal thefollowings:· Response in X direction: Table 2 shows that

modes 1, 4, 19, 20 are the structure modeswith high effective mass. Mode 1, with themaximum effective mass, is the global modeof the structure in the X direction. At thismode, the complete structure vibratesaround the joint to the launcher in the Xdirection. Mode 4 is related to the boom

vibration on the middle plate, and modes 19and 20 are related to the vibration ofelectronic box on the base plate. Because ofhigher mass, the boom and electronic boxproduce the maximum effect on the modalbehavior of the satellite structure.

· Response in Y direction: modes 2 and 47are the modes with high effective mass ofthe structure. Mode 2, with highest effectivemode, is the global mode of the structure inY direction. Mode 47 is related to the boomvibration on the middle plate.

· Response in Z direction: modes 11, 3, 61are the structure modes with high effectivemass. Mode 3 is the main structure mode inZ direction. At this mode, the middle and topplates, which are joined to each other by thestructural plates from the boom cover andthe boom, vibrate together. Mode 11 iscaused by deformation of the lower moduleunder the weight of the upper module (mainlybattery packages and the boom). Mode 61

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is due to the local modes of the lateral plates,which is described in detail in Ref. [9].

Figures 3 through 5 show that thesuperposition method takes into account the mainstructure modes.

Consideration of local responses from Nastran-Patran software reveal the followings:´ Response at the solar panels( at the joint to

the top plate): In X and Y directions, behaviorof the plates is similar to behavior of thecenter of gravity (with a smaller amplificationfactor). However, in Z direction, due todeformation of the lower module under theweight of the upper module, the solar panelsexperience large deformation at mode 11,which results in appreciable difference in theequivalent rms in this direction, which is infact because of local behavior of the solarpanels.

´ Response at the upper module stiffeners:Here again, a behavior similar to the centerof gravity can be seen in X and Y directions.However, in Z direction, due to position of

Table 3: Power SpectralDensity (PSD) of input in all

the three directions

Freq (Hz) PSD X (g²/Hz)

20 0.00850 0.02125 0.22135 0.22145 0.16200 0.162000 0.0325000 0.032

Table 4: Results at different points and center of gravity

rms in X direction rms in Y direction rms in Z direction

Response of center of gravity 6.7 6.7 8.8Location of boom base 8.39 9.64 8.66Center of solar panel 5.11 5.87 13.57Upper module stiffener 5.37 5.82 13.18

Response at the boom baseConsideration of response at the boom

base indicates that behavior of this point providesa good approximation for behavior of the satelliteat the center of gravity.

The analyses results show that behaviorof center of gravity for this structure provides a goodapproximation for the overall behavior of thestructure. However, slight local differences for thisstructure (such as difference at the solar panels)must be considered at the critical design stage, to

determine the applied loads on the installedcomponents such as the satellite solar panels.These loads are the initial interface compatibilityDocument (ICD) between the satellite manufacturerand the producer of satellite equipment.

The above analyses show that at theconceptual design stage of small satellite structures,the superposition method can be used as a quickand simple method for selection of general andpreliminary specifications of the satellite structure.At the later design stages, especially at the critical

the upper module stiffeners (these stiffeners,in the form of four pillars at the four corners,join the middle plate to the upper plate) thelower module experiences a considerabledisplacement under the weight of the uppermodule. This produces an appreciabledifference between the response of thecenter of gravity and the equivalent rms in Zdirection.

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design stage, the results will not be radically differentfrom the primary results.

Additionally, the analyses results show thata comparison of responses for the two methodscan be used to determine suitable position forinstallation of the accelerometers for showing thefrequency response (FRF) of the structure. For thestructure under consideration, the results alsoindicate that the boom base is positioned at asuitable point on the structure. The test results andmodel updating at the later stages validate theseresults [10].

It is seen that responses of each node,derived from superposition technique and finiteelement method, are in good agreement. A surveythrough the mode shapes shows that majordifferences occur when the nodes behave as neutralmodes at the specified frequency. The effect oflocalization of response can follow specially at highermodes. These fluctuations show that the selectednode has its local mode shapes, which can varydue to the mesh distribution and element selection.And these results are directly related to the locationof measurement points. The overall COG behaviorof structure can be calculated using this simplifiedtechnique.

CONCLUSIONS

The mode superposition techniques areapplied to calculate the frequency response ofsatellite structure. The effective mode selectioncriteria are applied based on the percentage ofmodal effective mass to extract the optimumselection criterion. The results are compared withfinite element analysis of the structure in selectedregion. It is seen that the results of superpositiontechnique are close to those calculating point thatare placed near the center of gravity of structure.It means that the superposition technique canrepresent the overall behavior of structure in CoG.These results can be applied as an equivalentstatic load due to random excitation to thestructures and can be reported as the overallbehavior of structure to the launcher that canconsider the small satellite as a concentratedmass with overall behavior like what is derivedfrom the superposition technique.

The results can also be used forapproximation of applied loads on the satellitestructure at early stages of design and selection ofsuitable locations for installation of sensors forobtaining the frequency response function of thestructure at the testing phase. For bigger structures,the local behavior can cause strong forces on thestructural elements. Therefore, behavior of centerof gravity will be a weaker approximation.

REFERENCES

1. Safarin, T.P., “Spacecraft Structures and

Mechanisms from Concept to launch”,

Microcosm, Inc., (1997).

2. Miles, G.W., “On Structural Fatigue under

Random Loading”, J. Aeronautical Sciences,753 (1954).

3. Plesseria, J.Y, Rochus, P. and Defise, J.M.,

“Effective modal masses”, 5th International

Congress of Theoretical and Applicational

Mechanics, France (2000).

4. Microsoft NASTRAN Theoretical Manual

(2005).

5. De Silva, “Vibration Fundamentals and

practice”, CRC Press (1999).

6. Wijker, J., “Mechanical Vibrations in

Spacecraft Design”, Springer (2004).

7. Emami, H., Farhani, F. and Safarabadi, M.,

“Influence of modal effective mass

distribution on the static and dynamic

behavior of a satellite structure under base

excitations”, Material Science ResearchIndia, 5(2): 209-218 (2008).

52 Emami & Farhani, Mat. Sci. Res. India., Vol. 6(1), 41-52 (2009)

8. Guyan, J., “Reduction of stiffness and mass

matrices”, AIAA J., 3: 380 (1965).

9. “Cosmos Launch System”, Payload User’s

Manual 2.1 (1999).

10. Safarabadi, M. and Emami, H., “Study on the

validity of updating of finite element for a

micro satellite structure using random

vibration tests”, Internatio