DYNAMICS OF A STRONGLY NONLINEAR SPACECRAFT STRUCTUREPART I: EXPERIMENTAL IDENTIFICATION

J.P. Noel, L. Renson, and G. Kerschen

Space Structures and Systems LabAerospace and Mechanical Engineering Department, University of Liege

1, Chemin des chevreuils (B52/3), 4000, Liege, Belgiumjp.noel, l.renson, g.kerschen@ulg.ac.be

ABSTRACT

The present paper addresses the identification of a real-life spacecraft structure possessing a strongly nonlinearcomponent with multiple mechanical stops. The com-plete identification procedure, from nonlinearity detec-tion and characterization to parameter estimation, is car-ried out based upon experimental sine-sweep data col-lected during a classical spacecraft qualification cam-paign.

Key words: Spacecraft structure; experimental data; non-linear system identification; modal interactions.

1. INTRODUCTION

Nonlinear system identification is a challenging task inview of the complexity and wide variety of nonlinearphenomena. Significant progress has been enjoyed dur-ing the last fifteen years or so [KWVG06] and, to date,multi-degree-of-freedom lumped-parameter systems andsimple continuous structures with localized nonlinearitiesare within reach. The identification of weak nonlineari-ties in more complex systems was also addressed in therecent past. The identification of large-scale structureswith multiple, and possibly strongly, nonlinear compo-nents nevertheless remains a distinct challenge and con-centrates current research efforts.

The present paper addresses the identification of theSmallSat spacecraft developed by EADS-Astrium, whichpossesses a vibration isolation device with multiple me-chanical stops. The complete identification procedure,from nonlinearity detection and characterization to pa-rameter estimation [KWVG06], will be achieved basedupon experimental sine-sweep data collected during aclassical spacecraft qualification campaign. Throughoutthe paper, the combined use of analysis techniques willbring different perspectives to the dynamics. Specifically,the spacecraft will be shown to exhibit particularly inter-esting nonlinear behaviors, including jumps and modal

interactions. Specific attention will be devoted to nonlin-ear modal interactions as their experimental evidence inthe case of a complex, real-life structure is an importantcontribution of this work. Note that a thorough numericalstudy of the SmallSat satellite dynamics is carried out ina companion paper [RNK14].

2. THE SMALLSAT SPACECRAFT STRUCTURE

The SmallSat structure was conceived by EADS-Astriumas a low-cost platform for small satellites in low earth or-bits. It is a monocoque tube structure which is 1.2min height and 1m in width. It is composed of eight flatfaces for equipment mounting purposes, creating an oc-tagon shape, as shown in Fig. 1. The top floor is an 1-m2 sandwich aluminum panel. The interface between thespacecraft and the launch vehicle is achieved via four alu-minum brackets located around cut-outs at the base of thestructure. The total mass including the interface bracketsis around 64kg.

The spacecraft structure supports a dummy telescopemounted on a baseplate through a tripod; its mass isaround 140kg. The dummy telescope plate is con-nected to the SmallSat top floor by three shock attenua-tors, termed shock attenuation systems for spacecraft andadaptor (SASSAs). Besides, as depicted in Fig. 2 (a),a support bracket connects to one of the eight walls theso-called wheel elastomer mounting system (WEMS) de-vice which is loaded with an 8-kg dummy inertia wheel.The WEMS device acts as a mechanical filter which mit-igates high-frequency disturbances coming from the in-ertia wheel through the presence of a soft elastomericinterface between its mobile part,i.e. the inertia wheeland a supporting metallic cross, and its fixed part,i.e.the bracket and by extension the spacecraft. Moreover,the WEMS incorporates eight mechanical stops, coveredwith a thin layer of elastomer, and designed to limitthe axial and lateral motions of the inertia wheel duringlaunch, which gives rise to strongly nonlinear dynamicalphenomena.

Fig. 2 (b) presents a simplified, yet relevant, modeling

Dummyinertiawheel

SASSAdevices

Dummytelescope

WEMSdevice

Main structure

Figure 1: SmallSat spacecraft equipped with an inertiawheel supported by the WEMS device and a dummy tele-scope connected to the main structure by the SASSA iso-lators.

of the WEMS device where the inertia wheel, owing toits important rigidity, is seen as a point mass. The fournonlinear connections (NCs) between the WEMS mo-bile and fixed parts are labeled NC 1 – 4. Each NCpossesses a trilinear spring in the axial direction (elas-tomer in traction/compression plus two stops), a bilinearspring in the radial direction (elastomer in shear plus onestop), and a linear spring in the third direction (elastomerin shear). The stiffness and damping properties of theWEMS were estimated during experiments carried out byEADS-Astrium at subsystem level (see Table 1), and willserve as reference values in this study. For confidential-ity, stiffness coefficients and clearances are given throughadimensionalised quantities.

Lateral Axial ZX and Y

Stiff. coeff. of the elast. plots 2 8Stiff. coeff. of the mech. stops 40 100

Clearance 2 1.5Damp. coeff. of the 37 63elast. plots (Ns/m)

Table 1: Reference stiffness and damping properties ofthe WEMS device estimated during experiments carriedout by EADS-Astrium at subsystem level.

Low-level random data were acquired throughout thetest campaign, specifically between each qualificationrun, to monitor the integrity of the structure. This wasperformed considering axial white-noise excitations fil-tered in 5 – 100Hz and driven via a base accelera-tion of 0.001g2/Hz. The low-level time series can beexploited to identify the linear modal properties of thespacecraft, utilizing transmissibility functions (TFs) as

no force measurement was available at the shaker-to-structure interface. The identification was carried outusing a frequency-domain subspace identification algo-rithm. The resulting estimates of the resonance frequen-cies and damping ratios of the spacecraft are given in Ta-ble 2.

The actual qualification test campaign consisted ofswept-sine base excitations applied to the spacecraft fordifferent amplitude levels, sweep rates and directions.Two specific data sets measured under 0.6g and 1g axialloadings and for positive sweep rates of 2 and 4 octavesper minute, respectively, are exploited in the present workfor nonlinear system identification. For conciseness, theiranalysis is focused in the next sections on the frequencyrange between 5 and 15Hz, i.e. the vicinity of the firstmode of vibration of the structure. The associated space-craft motion is depicted in Fig. 3 through the modal co-ordinates of the inertia wheel and telescope in the X, Yand Z directions. This motion consists mainly in a swingoscillation of the inertia wheel around Y-axis.

Mode Frequency (Hz) Damping ratio (%)

1 8.19 4.362 20.18 5.213 22.45 6.764 34.30 5.035 43.16 2.766 45.99 3.727 55.71 3.668 64.60 4.789 88.24 2.89

Table 2: Linear resonance frequencies and damping ra-tios estimated using a frequency-domain subspace iden-tification algorithm applied to low-level random data(0.001g2/Hz).

IW−X IW−Y IW−Z Tel−X Tel−Y Tel−Z

−1

−0.5

0

0.5

1

Modal coord

inate

Degree of freedom

Figure 3: First mode of vibration of the spacecraft de-scribed through the modal coordinates of the dummy in-ertia wheel and telescope in the X, Y and Z directions.

(a)X

Z

SmallSatInertia wheel

Bracket

Metalliccross

Filteringelastomer plot

Mechanicalstop

(b)

NC 4

NC 3

NC 2

NC 1

XY

Z

Inertiawheel

Figure 2: WEMS device. (a) Detailed description of the WEMS components; (b) simplified modeling of the WEMSmobile part considering the inertia wheel as a point mass. The linear and nonlinear connections between the WEMSmobile and fixed parts are signaled by squares and circles, respectively.

3. DETECTION OF NONLINEARITY

Nonlinearity detection is the first step of the identificationprocess [KWVG06], and basically boils down to seekingdepartures from linear theory predictions. In this regard,stepped- and swept-sine excitations are particularly con-venient because, if linear, the structure is known to gen-erate a pure sine wave in output, and distortions may bedetected without requiring complicated post-processing.

3.1. Envelope-based analysis of the raw time series

Nonlinear distortions in response to sine excitations cansometimes be such that a mere visual inspection of theraw time series is sufficient to reveal nonlinear behavior.To this end, the axial relative displacements across NC 1measured at 0.6g and 1g are plotted in Fig. 4 (a – b),respectively. Note that the measured accelerations wereintegrated twice using the trapezium rule and then high-pass filtered to obtain displacement signals. For confi-dentiality, relative displacements and velocities are adi-mensionalised throughout the paper.

The first observation is the absence of proportionality be-tween the time responses in Fig. 4 (a – b). This is espe-cially visible for negative displacements where the max-imum amplitude reached at 0.6g and 1g is almost un-changed. This violates the principle of superposition, acornerstone of the linear theory. The location of the res-onance in amplitude in the two graphs can also be seento be shifted towards higher frequencies, from 8.3 to 9Hz as the level is increased from 0.6 to 1g. One furtherremarks the clear skewness and nonsmoothness of the en-

velope of oscillations in Fig. 4 (b), which exhibits a sud-den transition from large to small amplitudes of vibration,referred to as a jump phenomenon. This envelope alsopresents a significant asymmetry entailing larger ampli-tudes of motion in positive displacement, and a disconti-nuity in slope for negative displacements around 7.5Hz.

By contrast, the envelope of response at 0.6g shows noevidence of nonlinear distortion. However, analyzing theresponse in the vicinity of resonance,i.e. in the 8.1 – 8.4Hz interval, as presented in Fig. 4 (c), highlights the pres-ence of harmonics in the time series. A similar inspectionat 1g, depicted in Fig. 4 (d) in 8.4 – 8.7Hz, reveals muchmore significant harmonics and a limitation of the ampli-tude of motion in negative displacement resulting in theaforementioned asymmetry of the response.

4. CHARACTERIZATION OF NONLINEARITY

Nonlinearity characterization is the second step of theidentification process, and amounts to selecting appropri-ate functional forms to represent the nonlinearities in thesystem. Characterization is of paramount importance, asthe success of the third step of the process,i.e. the esti-mation of model parameters, is conditional upon a preciseunderstanding of the nonlinear mechanisms involved. Itis also a very challenging step because the physical phe-nomena that entail nonlinearity are numerous and mayresult in plethora of dynamic behaviors.

5 7 9 11 13 15−2

−1

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1

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Sweep frequency (Hz)

Re

lative

dis

pla

ce

me

nt

at

NC

1

(a)

5 7 9 11 13 15−2

−1

0

1

2

Sweep frequency (Hz)

Re

lative

dis

pla

ce

me

nt

at

NC

1

(b)

8.1 8.2 8.3 8.4−2

−1

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1

2

Sweep frequency (Hz)

Re

lative

dis

pla

ce

me

nt

at

NC

1

(c)

8.4 8.5 8.6 8.7−2

−1

0

1

2

Sweep frequency (Hz)

Rela

tive d

ispla

cem

ent at N

C 1

(d)

Figure 4: Nonlinearity detection at 0.6g (left column) and 1g (right column). (a – b) Envelope-based analysis; (c – d)close-up of the displacement signals.

4.1. Restoring force surface plots

The restoring force surface (RFS) method [KWVG06]serves commonly as a parameter estimation technique, asin Section 5 of the present paper. This section introducesan unconventional use of the RFS method for nonlin-earity characterization purposes, relying exclusively onmeasured signals. The starting point is Newton’s secondlaw of dynamics written for a specific degree of freedom(DOF) located next to a nonlinear structural component,namely

N∑

n=1

mi,n qn + fi(q, q) = pi (1)

wherei is the DOF of interest,N the number of DOFs inthe system,mi,j the mass matrix elements,q, q andq thedisplacement, velocity and acceleration vectors, respec-tively, f the restoring force vector encompassing elasticand dissipative effects, andp the external force vector.The key idea of the approach is to discard in Eq. (1) allthe inertia and restoring force contributions that are not

related to the nonlinear component, as they are generallyeither unknown,e.g. the coupling inertia coefficients, ornot measured,e.g. the rotational DOFs. If we denoteby j another measured DOF located across the nonlinearconnection, Eq. (1) is therefore approximated by

mi,i qi + fi(qi − qj , qi − qj) ≈ pi. (2)

If no force is applied to DOFi, a simple rearrangementleads to

fi(qi − qj , qi − qj) ≈ −mi,i qi. (3)

Eq. (3) shows that the restoring force of the nonlinearconnection is approximately proportional to the acceler-ation at DOFi. Hence, by simply representing the ac-celeration signal, with a negative sign, measured at oneside of the nonlinear connection as a function of the rela-tive displacement and velocity across this connection, thenonlinearities can be conveniently visualized, and an ad-equate mathematical model for their description can thenbe selected.

−2 −1 0 1 2−80

−40

0

40

80

Relative displacement at NC 1

− A

ccele

ration a

t N

C 1

(m

/s2)

(a)

Sweep frequency (Hz)

Insta

nta

ne

ou

s f

req

ue

ncy (

Hz)

5 7 9 11 13 15

20

40

60

80

100

Am

plit

ud

e (

dB

)

−200

−160

−120

−80

(b)

Figure 5: Nonlinearity characterization of the WEMS device at 1 g using (a) the restoring force surface method and (b)the wavelet transform.

To visualize the elastic nonlinearities of the WEMS de-vice, a cross section along the axis where the velocity iszero of the restoring force surface defined by the triplets(qi,k− qj,k, qi,k− qj,k, −qi,k), wherek refers to thek-thsampled instant, can be drawn. Fig. 5 (a) shows the plotcorresponding to NC 1 at 1g. This figure is particularlyuseful as it reaffirms the nonsmooth and asymmetric na-ture of the nonlinearities in the system, and the estimationof the – Z clearance at around 1. It also reveals the acti-vation of the + Z stop, beyond a relative displacement ofabout 1.5.

4.2. Time-frequency analysis using the wavelettransform

One of the most suitable tools for interpreting harmon-ics generated by nonlinear systems in response to swept-sine excitations is the wavelet transform (WT) [Sta00].The wavelet amplitude of the relative displacement ofFig. 4 (b) is displayed in logarithmic scaling in Fig. 5 (b).The appearance of wideband frequency componentsaround 7.5Hz, including even harmonics, confirms theactivation of a nonsmooth nonlinearity in the neighbor-hood of the resonance and the existence of an asymme-try in the system. The disappearance of the widebandcontent is seen to coincide closely with the jump phe-nomenon observed in Fig. 4 (b). One should also pointout that impurities in the input sine wave turn into weakharmonics visible throughout the spectrum, but hence notattributable to nonlinearity. Similarly, electrical noise isresponsible for a polluting frequency line around 50Hz.

In summary, the nonlinearity characterization step revealsthat an accurate representation of the WEMS nonlinearbehavior should account for combined nonsmooth andasymmetric effects. This leads us to select a trilinearmodel with dissimilar clearances for the nonlinearity. Nocharacterization of damping was attempted in this sec-

tion as the scope of the paper is focused on the identifica-tion of the nonlinear dynamics introduced by the WEMSmechanical stops. One therefore opts for a simple lin-ear damping model for the elastomer components of theWEMS.

4.3. Evidence of nonlinear modal interactions

The WT can evidence a salient feature of nonlin-ear systems that has no counterpart in linear the-ory, namely modal interactions between well-separatedmodes [KPGV09]. To reveal nonlinear modal interac-tions in the SmallSat dynamics, Fig. 6 (a) depicts thewavelet amplitude of the acceleration measured at NC4 in the Z direction over 5 – 35Hz. Compared to thewavelet represented in Fig. 5 (b), a linear scale is usedherein to focus on the most significant frequency com-ponents in the time series. The excitation frequency isclearly seen throughout the wavelet, but higher harmoniccomponents of at least comparable amplitude are also vis-ible. In particular, a significant level of response, encir-cled in Fig. 6 (a), is observed around 60Hz for sweepfrequencies just below 30Hz. This corresponds to a2:1 interaction between two internally resonant modes ofthe structure, namely mode 3, which involves an out-of-phase motion of the inertia wheel and the WEMS bracket,and mode 7, which consists in an axial motion of the tele-scope supporting panel. The existence of a 2:1 interactionbetween modes 3 and 7 is confirmed in Fig. 6 (b) wherethe raw acceleration signal measured at the center of theinstrument panel is plotted at 0.1g and 1g. A high ampli-tude response at 1g is observed between 20 and 30Hz,which can be confidently attributed to a nonlinear reso-nance as no linear mode of the panel is located in this in-terval. One also remarks the presence of two resonancesaround 46 and 56Hz, as predicted by the linear modalanalysis carried out in Section 2. At the 0.1g excitationlevel for which the satellite behaves linearly, there is no

Sweep frequency (Hz)

5 10 15 20 25 30 350

20

40

60

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Am

plit

ud

e (

m2/H

z2)

0

20

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120

4:1

5:2

2:1

(a)

5 10 20 30 40 50 60 70−100

−50

0

50

100

Sweep frequency (Hz)

Acce

lera

tio

n (

m/s

2)

(b)

Figure 6: (a) Evidence of nonlinear modal interactions using the WT calculated at NC 4 in the Z direction, and presentedin 5 – 35Hz in linear scaling. A 2:1 modal interaction between modes 3 and 7 is encircled; (b) confirmation of theexistence of the 2:1 modal interaction through the raw acceleration measured at the center of the instrument panel at 0.1g (in blue) and 1g (in black).

sign of the 2:1 modal interaction, proving that it is an in-herently nonlinear phenomenon activated for sufficientlylarge energies.

It should also be stressed that this 2:1 modal interactionmay jeopardize the integrity of the structure as it is ac-companied by an energy transfer from a local mode ofthe spacecraft with low effective mass,i.e. mode 3, to aglobal mode with high effective mass,i.e. mode 7. Inaddition, the time series at 1g in Fig. 6 (b) shows thatthe nonlinear resonance involving the instrument panel isassociated with larger accelerations (i.e., 100m/s2) thanthe linear resonance of the panel (i.e., 80m/s2). This im-plies that important, and potentially dangerous, dynami-cal phenomena can be missed when ignoring nonlinear-ity.

5. PARAMETER ESTIMATION IN THE PRES-ENCE OF NONLINEARITY

Based on the choice made in Section 4 to use trilinearfunctional forms to characterize the WEMS elastic be-havior together with linear damping properties, the laststep toward the establishment of a nonlinear model withgood predictive capabilities is the estimation of the asso-ciated parameters,i.e. the clearances, stiffness and damp-ing coefficients. It turns out from a survey of the technicalliterature [KWVG06] that the RFS method is one of theonly approaches compatible with unmeasured base-sineexcitations.

Though it was shown to extend to multi-DOF systems,the utilization of the RFS method is in general limited tosmall-scale structures, since the starting point of the ap-proach is a rigorous writing of Newton’s second law of

dynamics. In Ref. [NRK14], it is however demonstratedthat the equations of motion of the WEMS mobile partcan be formulated explicitly by asserting that it behavesas a rigid body. The dynamics of a rigid body is knownto obey 6 scalar equations, namely 3 equations describ-ing the translation of its center of gravity, and 3 equationsgoverning the rotation of the body around the center ofgravity. One herein concentrates on the translation alongZ-axis of the center of gravity of WEMS mobile part. Un-der the rigidity assumption, the corresponding equationof motion writes

mz1 + z2

2+ fNC 1(z1)+ c1 z1+ fNC 2(z2)+ c2 z2 = 0

(4)wherem is the mass of the WEMS mobile part estimatedat 8.75kg, fNC 1 andfNC 2 the trilinear stiffness forces,c1 andc2 the linear damping coefficients of the NC 1 andNC 2 elastomer plots, respectively, and where the contri-butions related to NC 3 and NC 4 are neglected as theyinvolve considerably lower displacements and velocities.This results from the swing motion of the WEMS aroundY-axis observed in the frequency band of interest (seeFig. 3 in Section 2).

The restoring force surfaces constructed via Eq. (4) at NC1 and NC 2 given the triplets (z1, z1, fNC 1 + c1 z1) and(z2, z2, fNC 2 + c2 z2), respectively, can be fitted using atrilinear model in stiffness and a linear model in damping.Curve-fitting results are given in Table 3, and were com-puted in the 8.5 – 9Hz interval in which the first reso-nance of the system is located. The damping coefficientsof the elastomer plots are found to be much larger thantheir reference value of 63Ns/m (see Table 1), but areprone to significant uncertainty as they were computedfrom a limited number of low-displacement samples. Asan illustration, the stiffness curve extracted as a cross sec-tion of the corresponding restoring force surface at NC 1

Reference value NC 1 NC 2

Linear damping coefficientc (Ns/m) 63 218.29 147.75Linear stiffness coefficientk 8 8.30 9.21Neg. clearance 1.5 1.01 0.84Pos. clearance 1.5 1.55 1.62Neg. nonlinear stiffness coefficient 100 118.07 116.73Pos. nonlinear stiffness coefficient 100 79.40 88.41

Table 3: Damping coefficients, stiffness coefficients and clearances of NC 1 and NC 2 estimated using the RFS methodand compared with their reference values.

is plotted in Fig. 7, together with the fitted trilinear model.The calculated coefficients show that the stiffnesses ofthe elastomer plots and mechanical stops match well theirreference values of 8 and 100 (see Table 1), respectively,considering that no asymmetry was introduced in the ref-erence model. Impacts are also found to be comparativelysofter for positive displacements.

−2 −1 0 1 2−800

−400

0

400

800

Relative displacement at NC 1

Resto

ring forc

e a

t N

C 1

(N

)

Figure 7: WEMS nonlinear stiffness curve constructedbased upon Eq. (4) (in black) and compared with the fittedtrilinear model (in blue) at NC 1.

6. CONCLUSION

In this paper, the experimental identification of a real-life spacecraft structure exhibiting strongly nonlinear dy-namics due to multiple mechanical stops was achievedbased on sine-sweep data. The complete progressionthrough nonlinearity detection, characterization and pa-rameter estimation was carried out by means of severalexisting analysis techniques, yielding an accurate estima-tion of the clearances and nonlinear stiffness properties ofthe nonlinear components. Moreover, this paper demon-strated that the complex dynamics that can be obtainedduring numerical simulations of nonlinear systems withlow dimensionality can also be observed in experimentalconditions commonly endured by engineering structuresin industry. This includes intrinsically nonlinear, and po-tentially dangerous, phenomena such as jumps and modalinteractions.

ACKNOWLEDGMENTS

This paper was prepared in the framework of theEuropean Space Agency (ESA) Technology ResearchProgramme study “Advancement of Mechanical Veri-fication Methods for Non-linear Spacecraft Structures(NOLISS)” (ESA contract No.21359/08/NL/SFe). Ex-perimental data were measured by EADS-Astrium andLMS International. The authors also thank Astrium SASfor sharing information about the SmallSat spacecraft.The authors J.P. Noel and L. Renson are ResearchFellows (FRIA fellowship) of theFonds de la RechercheScientifique – FNRSwhich is finally gratefully acknowl-edged.

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[KWVG06] G. Kerschen, K. Worden, A.F. Vakakis, andJ.C. Golinval. Past, present and future ofnonlinear system identification in structuraldynamics. Mech. Syst. Signal Pr., 20:505–592, 2006.

[NRK14] J.P. Noel, L. Renson, and G. Ker-schen. Complex dynamics of a nonlinearaerospace structure: Experimental identifi-cation and modal interactions.J. Sound Vib.,333:2588–2607, 2014.

[RNK14] L. Renson, J.P. Noel, and G. Kerschen.Dynamics of a strongly nonlinear space-craft structure. Part II: Modal analysis. InProc. of the Eur. Conf. on Spacecraft Struc-tures, Materials and Environmental Testing,Braunschweig, Germany, 2014.

[Sta00] W. J. Staszewski. Analysis of non-linearsystems using wavelets.P. I. Mech. Eng. B-J. Eng., 214(11):1339–1353, 2000.