Eindhoven University of Technology

MASTER

Nonlinear dynamics of a MDOF solar array system

van Liempt, F.P.H.

Award date:1996

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96-CMC-R1582 Nonlinear Dynamics of a MDOF Solar Array System

4 december 1996

F.P.H. van Liempt

Afstudeerrapport van F .P.H. van LiemptWFW-rapportnr . : 96.159December 1996

Afstudeer hoogleraar : Prof.dr .ir. D.H. van Campen (TUE)Afstudeer begeleider: Dr.Ir. R.H.B. Fey (TNO)

Dr.Ir. A. de Kraker (TUE)Ir. J .J. Wijker (Fokker Space B.V., TUD)

Projectnr . :62385430Goedgekeurd : Dr .ir . R.H.B. Fey

Pagina's : 38 -f 1,3Bijlagen :42k7

Nonlinear Dynamics of a MDOF Solar Array System

F.P.H. van LiemptDecember 1996

Thesis for the engineering degreeF.P.H. van Liempt, id. 321650WFW-report id. WFW96 .159

Professor : Prof. Dr. Ir. D .H. van Campen, TUECoaches : Dr. Ir. R.H.B. Fey, TNO

Dr. Ir. A. de Kraker, TUEIr. J .J. Wijker, TUD/Fokker Space b.v .

Eindhoven University of TechnologyDepartment of Mechanical EngineeringDivision of Fundamental Mechanics

I would like to thank my parents who were a tremendous support during my study, and Tina .Further, I would like to thank all people who reviewed this report, especially Rob, and TNOBuiling and Construction Research for providing the means and support for this research .

I

11

Samenvatting

In de werktuigbouwkundige praktijk worden complexe mechanische systemen vaak geanaly-seerd met behulp van eindige elementen modellen met veel vrijheidsgraden voor een voldoendenauwkeurige beschrijving van het dynamisch gedrag. Zogenaamde niet-lineariteiten in mech-anische systemen hebben, indien aan wezig, vaak een lokaal karakter, in die zin dat ze slechtsgerelateerd zijn aan een gering aantal vrijheidsgraden . De aanwezigheid van deze lokale niet-lineariteiten kan echter van grote invloed zijn op het dynamisch gedrag van het systeem inzijn geheel .

In dit rapport beschouwen we een zonnepanelenconstructie . Tijdens de lancering van eensatelliet wordt de zonnepanelenconstructie in opgevouwen toestand tegen de satelliet aangek-lemd om ruimte te sparen . Tevens is de constructie in deze toestand beter bestand tegen deperiodieke excitatie zoals die optreedt tijdens de lancering . Bij bepaalde excitatie frequentieskan de mogelijkheid ontstaan dat de panelen tegen elkaar slaan . Om dit te voorkomen wor-den op goed gekozen plaatsen zogenaamde snubbers gemonteerd : dit zijn eenzijdige lineaireveren, die dienen als een elastische aanslag . Het belangrijkste voordeel van deze oplossing isdat er geen speciale aanpassingen nodig zijn om de zonnepanelenconstructie te ontvouwen .Echter, toepassing van snubbers in een bij benadering lineaire constructie leidt tot een mecha-nisch systeem met lokale niet-lineariteiten. De eindige elementen methode, gekoppeld aan eenreductie-module en numerieke algoritmen voor het analyseren van niet-lineaire dynamischesystemen wordt gebruikt bij de bepaling van het dynamisch gedrag van dit systeem .

Fokker Space b .v., de fabrikant van de zonnepanelenconstructie, is geïnteresseerd in het langetermijn gedrag van deze constructie voor het frequentiebereik van 20 - 100 Hz .

Niet-lineaire dynamische analyse is uitgevoerd op basis van een realistisch 3D eindige el-ementen model van de zonnepanelenconstructie in opgevouwen toestand (±6000 vrijheids-graden) . De belangrijkste onderdelen zijn :

0 twee zonnepanelen ;. twee zogenaamde holddown and release systemen . Deze houden de panelen tijdens

de lancering in opgevouwen toestand tegen de satelliet aangeklemd, en waarborgentevens een constante afstand tussen de panelen . Omdat de panelen flexibel zijn geldtdit alleen voor de directe omgeving van de holddowns ;

0 een yoke. Deze arm is de enige verbinding tussen de satelliet en de operationele zon-nepanelenconstructie. De yoke kan de ontvouwde zonnepanelenconstructie in een op-timale stand ten opzichte van de zon draaien . Tijdens de lancering is de yoke langs depanelen gevouwen ;

0 een aantal scharnieren, nodig voor het ontvouwen van de zonnepanelenconstructie (ditproces wordt niet beschouwd) ;

. vier snubbers (lokale niet-lineariteiten), geplaatst tussen de panelen .

in

Het aantal vrijheidsgraden van het lineaire deel van de constructie (hele constructie zon-der snubbers) wordt gereduceerd met behulp van een component mode sythese methode,die gebaseerd is op free-interface eigenmodes en residuele flexibiliteits modes . Koppelingvan de vier snubbers aan het gereduceerde lineaire model geeft een gereduceerd niet-lineairmodel. Dit heeft 14 vrijheidsgraden en is nauwkeurig binnen het beschouwde frequentie-interval . De basis van dit model wordt periodiek geëxciteerd met een constante versnellings-amplitude van 2g. Periodieke oplossingen worden berekend door het oplossen van een twee-punts-randvoorwaardenprobleem door toepassing van eindige differentie . Met een boog con-tinuërings methode (path following) wordt onderzocht hoe een periodieke oplossing verandertbij een verandering in de excitatiefrequentie . De lokale stabiliteit van een periodieke oplossingwordt bestudeerd met de theorie van Floquet .

De zonnepanelenconstructie blijkt erg "rijk" dynamisch gedrag te vertonen : 1/2, 1/3 en1/4 subharmonische oplossingen zijn gevonden, evenals quasi-periodiek en chaotisch gedrag .Vooral de tak met 1/2 subharmonische oplossingen is van belang voor de praktijk, omdatdeze bestaat in een erg groot deel van het beschouwde frequentiebereik (±75 - 100 Hz) . Dezetak heeft een maximum bij 88 Hz : de amplitude van de stabiele 1/2 subharmonische oplossingis een factor 10 groter dan de amplitude van de instabiele harmonische oplossing bij dezefrequentie .

Fokker wil weten of er lineaire sommen gedraaid kunnen worden, waarbij de snubbers doorstandaard lineaire veren worden vervangen, waarvan de resultaten in de buurt komen vande veel duurdere niet-lineaire dynamische analyse . Voor het hier beschouwde systeem is hetantwoord NEE. De verschillen in amplitude tussen drie lineaire analyses en de niet-lineaireanalyse zijn bij bepaalde excitatie frequenties minstens een factor 10 .

Tot slot : voor het niet-lineaire model is secondary Hopf bifurcatie van een stabiele 1/4 subhar-monische oplossing in een quasi-periodieke "torus" gevonden . Deze quasi-periodieke oplossin-gen geeft vier gesloten krommen in het Poincaré vlak .

iv

Summary

In engineering practice, complex mechanical systems often have to be analysed by meansof multi-degree-of-freedom (MDOF) models to obtain a sufficiently accurate description oftheir dynamic behaviour . Frequently, however, the nonlinearities in mechanical systems havea local character, which means that from a geometrical point of view they are only related toa limited number of degrees of freedom . The presence of local nonlinearities in a mechanicalsystem can, however, have important consequences for its overall dynamic behaviour .

In this report a solar array system under periodic excitation is considered . During the launchof a satellite, its solar array is in folded position to save space on behalf of the fairing, whereasit is exposed to high vibration levels . In this folded position, it is possible that the panelsstrike each other if the excitation is too severe . To prevent this, snubbers are mounted at wellchosen points of the structure . They act as elastic stops, and can be modelled as one-sidedlinear springs . The major advantage of this solution is that no special adjustments are neededfor unfolding, whereas the practical implementation is simple . Application of snubbers in theapproximately linear folded solar array structure leads to a mechanical system with localnoniinearities . For the dynamic analysis of complex mechanical systems the finite elementmethod is a well developed analysis tool . For mechanical systems with local nonlinearitiesthe finite element technique has been integrated with numerical tools for the analysis of non-linear dynamical systems to obtain a tool for efficient dynamic analysis of nonlinear MDOFengineering systems .

Fokker Space b.v., the manufacturer of the specific solar array system considered in thisreport, is interested in the long-term behaviour of this system under periodic excitation inthe frequency range 20 - 100 Hz .

The nonlinear analyses are performed on a realistic 3D finite element model of the solar arrayin folded position (±6000 dof). The major parts are :

. two solar panels ;

. two holddown and release systems . They clench the solar array to the body of thesatellite, and maintain a constant distance between the panels . This is only true inthe immediate vicinity of the holddown's, because the panels are flexible ;

. one yoke. This arm is the only connection between the solar array and the operationalsatellite . The yoke is able to rotate the spread-out solar array to an optimal positionwith respect to the sun . During the considered launch this arm is folded along thepanels ;

0 several joints, around which the panels can rotate during unfolding (this process isnot subject of investigation)

. four snubbers, i .e. four local nonlinearities, placed between the panels .

v

The original finite element model of the solar array system described above contains about6000 degrees of freedom . The number of degrees of freedom of the linear part of the solararray model, containing the satellite and the solar array except the snubbers, is reducedby applying a component mode synthesis technique, based on free-interface eigenmodes andresidual flexibility modes . Coupling of the four snubbers to the reduced structure results in anonlinear 14 dof model, which is accurate within the frequency range of interest . This modeli5 exposed to in phase periodic base excitation with an acceleration amplitude of 2g . Periodicsolutions are calculated by solving a two-point boundary value problem by applying a finitedifference method. How the periodic solution is influenced by a change in the excitationfrequency is investigated by applying a path following technique . The local stability of theperiodic solution is investigated using Floquet theory.

The system reveals very rich dynamic behaviour : 1/2, 1/3 and 1/4 subharmonic solutions arefound, as are quasi-periodic and chaotic behaviour . Especially the stable 1/2 subharmonicsolution branch is relevant in practice, because it spans such a wide frequency range (±75 -100 Hz) . The difference in amplitude between this solution and the accompanying unstableharmonic solution reaches a maximum at 88 Hz, where the amplitude of the 1/2 subharmonicsolution exceeds the amplitude of the corresponding harmonic solution by a factor 10 .

The solar array manufacturer is interested in the possibility of replacing the one-sided springsby two-sided (standard normal) springs . The analysis reveals that at an excitation frequencyof 88 Hz the relative difference between the amplitude of the 1/2 subharmonic solution forthe reduced nonlinear model and the harmonic solution for the modified linear model withtwo-sided springs remains of the same order of magnitude as the factor 10 mentioned above .

Finally it is remarked that for the nonlinear model bifurcation of a stable 1/4 subharmonicsolution, via a secondary Hopf bifurcation, into a quasi-periodic "torus" is found. This quasi-periodic solution shows four closed curves in a Poincaré section .

vi

Contents

1 Introduction 1

2 Theory 32.1 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 .2 Some nonlinear phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 .2 .1 Fundamental steady-state responses . . . . . . . . . . . . . . . . . . . 42 .2 .2 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . • • . . • . . 5

2 .3 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 .3 .1 Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 .2 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 .3 Definition of a Poincaré section . . . . . . . . . . . . . . . . . . . . . . 72.3 .4 Path following . . . . . . . . . . . . . . . . . . . . . . . . . . • • • • • 7

2.4 Local stability and local bifurcations . . . . . . . . . . . . . . . . . . . . . . . 8

3 System modelling 113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • • • . . 113.2 Model assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . • . • • • . 113.3 Finite element modelling of the linear part . . . . . . . . . . . . . . . . . . . . 11

3.3 .1 Eigenvalue analysis of the linear model . . . . . . . . . . . . . . . . . . 133.4 Reduction of the linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Completing the nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 .5 .1 Modelling of the snubbers . . . . . . . . . . . . . . . . . . . . . . . . . 173 .5 .2 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Linear dynamicanalysis 21

5 Nonlinear dynamic Analysis 255.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • • • . • 255 .2 Considerations on the solar array system . . . . . . . . . . . . . . . . . . . . . 255 .3 Weakly nonlinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 .4 Strongly nonlinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 .4 .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 .4 .2 Global discussion of system dynamics . . . . . . . . . . . . . . . . . . 275 .4 .3 Detailed discussion of the encountered phenomena . . . . . . . . . . . 285 .4 .4 Fourier transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 .5 Very strongly nonlinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Comparison nonlinear / linear 35

vii

7 Conclusions & Recommendations 377.1 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Case specific conclusions . . . . . . . . . . . . . . . . . . • • . • . . . . . . . . 377.3 Recommendations . . . . . . . . . . . . . . . . . . . . • • • • • . . . . . . . . 38

A Eigenmodes

B Linear dynamics

C Frequency responses, strongly nonlinear

D Demonstration internal excitation

E 1/17 Subharmonic solution

F Strongly nonlinear

G Frequency spectra

H Comparison nonlinear / linear

I Path following frequency ranges

J Finding a periodic solution : example

39

45

49

53

57

61

69

71

73

75

K Numerical data of the model 77K.1 Solar Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77K .2 Edgemembers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78K.3 Holddown systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

K .3.1 Holddown between satellite and panel 1 (lower panel) . . . . . . . . . 78K .3.2 Holddown between panel 1 and panel 2 . . . . . . . . . . . . . . . . . 78

K.4 Yoke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78K.5 Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

L User-force subroutines 81L.1 Definition external load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81L .2 Definition one-sided spring . . . . . . . . . . . . . . . . . . . . . . . • . . . . . 83

List of Figures

1 .1 Example of a solar array system . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.1 Schematic view of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Schematic view of the reduced model . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Frequency responses on the basis of linear dynamic analysis : k=0 Nm-1 . . . 234.2 Frequency responses for three stiffness values of the two-sided springs . . . . 24

5.1 Frequency response (14 dof), ksnnb = 1 .104 Nm-1 (weakly nonlinear) . . . . . 315.2 Frequency responses of ifcnod 1 and 6 (nonlinear) . . . . . . . . . . . . . . . . 325.3 Enlargement of the boxed area of fig . 5.2b . . . . . . . . . . . . . . . . . . . . 335.4 Unstable harmonic accompanying stable 1/3 subharmonic . . . . . . . . . . . 335.5 Quasi-periodic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.6 Comparison of frequency spectra of harmonic and chaotic response . . . . . . 34

6.1 Comparison of linear dynamic analyses and nonlinear dynamic analyses . . . 36

A.1 Free interface eigenmodes of the supported linear system . . . . . . . . . . . . 40A.2 Free interface eigenmodes of the supported linear system . (continued) . . . . 41A.3 Free interface eigenmodes of the supported linear system . (continued) . . . . 42A.4 Free interface eigenmodes of the supported linear system . (continued) . . . . 43A.5 Free interface eigenmodes of the supported linear system . (end) . . . . . . . . 44

B.1 Linear dynamic analyses for three spring stiffness values . . . . . . . . . . . . 47

CA Frequency responses of ifcnod 2 and 3, ksnub = 1 .10' Nm-1 . . . . . . . . . . 50C .2 Frequency responses of ifcnod 4 and 5, ksnub = 1• 105 Nm-1 . . . . . . . . . . . 51C .3 Frequency response of ifcnod 7, ksnub = 1-10' Nm-1 . . . . . . . . . . . . . . . 52

D .1 Ifcnod 5, feX = 20 .49 Hz, harmonic solution, nT = 2000 . . . . . . . . . . . . . 54D .2 Stable harmonic solution at feX = 20.49 Hz, nT = 2000 . . . . . . . . . . . . . . 55

E.1 Quasi-periodic behavior locked on a 1/17 subharmonic . . . . . . . . . . . . . 58E.2 Poincaré section of a stable harmonic solution . . . . . . . . . . . . . . . . . . 58E.3 Floquet multipliers for 34 .69 < feX < 34 .83 Hz . . . . . . . . . . . . . . . . . . 59E.4 Phase portraits of unstable harmonic and stable 1/2 subh . at 88 Hz . . . . . 59

F .1 Stable 1/4 subharmonic solution at fe7t = 96 .15 Hz . . . . . . . . . . . . . . . . 62F .2 Phase portraits of stable 1/4 subharmonic solutions . . . . . . . . . . . e . . 63F .3 Floquet multipliers of the 1/4 subharmonic solution branch . . . . . . . . . . 63

ix

F .4 Phase portraits of quasi-periodic solution . . . . . . . . . . . . . . . . . . . . 64F .5 Poincaré section of quasi-periodic behaviour . . . . . . . . . . . . . . . . . . . 65F.6 Simultaneous bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66F.7 Poincaré sections of chaotic solution at 99 .136 Hz . . . . . . . . . . . . . . . . 67F.8 Time histories and phase portraits of chaotic solution . . . . . . . . . . . . . 68

G.1 Frequency spectra of harmonic, quasi-periodic and chaotic response . . . . . . 70

H .1 Comparison of frequency responses of linear and nonlinear analyses . . . . . . 72

x

List of Tables

3.1 Structure of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Contraints at important point of the structure . . . . . . . . . . . . . . . . . 133.3 Eigenfrequencies before and after meshrefinement . . . . . . . . . . . . . . . . 143.4 Eigenfrequencies of the free linear component . . . . . . . . . . . . . . . . . . 153.5 Comparison of the reduced and unreduced system . . . . . . . . . . . . . . . 163 .6 Transition to reduced model . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . 16

5 .1 Values of a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 .2 Stability of the harmonic solution branch near chaos . . . . . . . . . . . . . . 30

B .1 Eigenfrequencies before and after meshrefinement . . . . . . . . . . . . . . . . 46B .2 Eigenfrequencies for 3 values of the linear spring stiffnesses . . . . . . . . . . 46

I.1 Start and terminal frequencies of path following efforts (ksnub=105 Nm-1) . . 74

xi

Xll

Chapter 1

Introduction

The assignment for this research comes from the Dutch space industry .

Fokker Space b .v. is, among other things, manufacturer of solar arrays which are connectedto satellites . The company is interested in de dynamic behaviour of a specific solar array,consisting of two solar panels . While mounted onto a satellite in folded position, these arrayssuffer from high vibrations during the launch . It is possible that the panel ends strike eachother if the excitation is too severe . To prevent this, snubbers are mounted at well chosenpoints of the structure . They act as elastic stops, and can be modelled as one-sided linearsprings. The major advantage of this solution is that no special adjustments are needed forunfolding, whereas the practical implementation is simple . Application of snubbers in theapproximately linear folded solar array structure leads to a nonlinear dynamic system of thetype 'dynamic system with local nonlinearities' .

Fokker is interested in the steady-state behaviour of this system in the frequency range 20 -100 Hz, under periodic excitation with a constant acceleration amplitude of 2g perpendicularto the surface of the panels . This behaviour is investigated using DIANAtm (release 6 .1, 1996),developed by TNO Building and Construction Research . DIANA offers a unique coupling of

finite element modelling, reduction techniques and nonlinear dynamics solution procedures .

(the latter are collected in module STRDYN : 'STRUCtural DYNamics') .

Wijckmans ('95)[15] studied on a similar subject before . He used a 2D beam model of onlythe so-called "hang over" of the panels . One snubber was placed between the panel ends . Thisreport handles the dynamic behaviour of a complete solar array system : nonlinear analysesare performed based on a realistic 3D finite element model of the solar array in folded position(±6000 dof). The major parts are :

. two solar panels ;9 two holddown and release systems . They clench the solar array to the body of the

satellite, and try to maintain a constant distance between the panels . This is onlytrue in the immediate vicinity of the holddowns, because the panels are flexible ;

. one yoke. This arm is the only connection between the solar array and the operationalsatellite. The yoke is able to rotate the spread-out solar array to an optimal positionwith respect to the sun . During the considered launch this arm is folded along thepanels ;

0 several joints, around which the panels can rotate during unfolding (this process is nosubject of investigation) ;

. four snubbers, i .e. four local nonlinearities, placed between the panels .

2

Advanced Rigid Solar Arra~

while mounted unto tile

~pamtaït in folded Positiotu

tN.e,,c array-, suffer highvibrations during take-off.Rubber snubbcn, an applied to

cednce damage risk

ll:o:rnr n ivkk., ..vu! JIsums) .

Introduction

Figure 1 .1 : Example of a solar array system . Note : The system considered in this reportconsists of only two, much larger solar panels, clenched to the satellite with 4 holddownsinstead of 6 as displayed here . Only the dynamics of the system in fully folded position aresubject of investigation .

Figure 1.1 gives an impression of a similar solar array system, consisting of seven solar panelsand a yoke in V- configuration -

The following questions are starting-points for this research :

. Are snubbers needed to avoid contact between the panels?

. What is the influence of snubbers on the overall dynamic behaviour? Do they have a'positive' influence, i .e. do they damp the response?

. Are there big differences between nonlinear dynamic analyses and linear dynamicanalyses with linearized snubbers, i .e. standard two-sided sprigs?

This report starts with a brief discussion on some nonlinear phenomena and solution proce-dures which are relevant for this report . Chapter 3 deals with the system modelling . First afinite element model of the linear part of the structure is made . The number of degrees of free-dom of the linear model is reduced using a component mode synthesis method . Coupling ofthe local nonlinearities (snubbers) to the reduced linear model leads to the reduced nonlinearmodel. In chapter 4 linear dynamic analyses are carried out . The snubbers in the model arereplaced by standard two-sided linear springs . Three cases are investigated : no springs, weaksprings, and stiff springs . The results of the actual nonlinear dynamic analyses are discussedin chapter 5 . The system reveals very rich dynamic behaviour : 1/2, 1/3 and 1/4 subharmonic,quasi-periodic and chaotic solutions are met . The results of the nonlinear dynamic analysesand the linear dynamic analyses are compared in chapter 6 . Finally, chapter 7 contains someconclusions and recommendations .

Chapter 2

Theory

2 .1 ReductionSince nonlinear dynamic analyses are very expensive, in terms of cpu-time, it often is ad-vantageous to reduce the number of dof of linear system parts, using a Component ModeSynthesis method. For an extensive discussion on this subject, one is referred to Fey ('92) [3] .

Here, the discussion of reduction is restricted to what is relevant for this report : the reductionof a nonlinear system which can be thought of some local nonlinearities, of type 'one-sidedspring', all of them coupled to one linear component .

The equations of motion of the linear component can be written as :

(2.1) Mi +Bx+Kx= .f

The linear model has n, dof, consequently M, B and K are all of size n, x n, . Column x,

length nx , is partitioned in a nB-column xB of boundary dof and a ni-column x , of unloadedinternal dof: x= [xB xÍ]t . For the nx column with loads holds : f = [fB OÍ]t

In order to reduce the number of degrees of freedom the pseudo-load method is used . In thismethod nonlinearities are treated as pseudo-loads, as they are placed on the right-hand sidein the equations of motion . The remaining linear left-hand side is, once, subject to reduction .

The idea is to describe the linear dynamics of a component with a Ritz-approximation of thedisplacement field : The displacement field of a linear component is a linear combination of itseigenmodes . Using a dynamic component mode set, it is possible to describe the components'displacement field for excitation frequencies up to a certain cut-off frequency fc including onlyeigenmodes up to fc. The dynamic component mode set T will be defined as :

(2.2) T = [41~G '0 4%]

with II) G the residual flexibility modes, V the rigid body modes and 4-h the kept free-interfaceeigenmodes . A residual flexibility mode is defined for each boundary (interface) dof and guar-antees unaffected static load behaviour of the reduced system model . The displacement fieldx is then approximated by

(2.3) x = T q, q=[qB qR qk]t

Using (2 .3) in (2.1) leads to the reduced component equations :

TtMTq+TtBTq+TtKTq =T`f

(2.4) M9 4+ Bq 4+ Kg q= fQ

Theory

The actual assembling of the reduced component models in order to derive the reduced linearsystem model is simply achieved by replacement of interface dof of the reduced componentqB in (2.3) by the dof of the reduced system z :

(2.5) q = [zt qR qk]t

It is this simple, because the reduced linear system consists only of one reduced component .

Coupling of the local nonlinearities to the interface dof of the reduced linear component leadsto the reduced nonlinear system. This is done by demanding equilibrium of the interfaceloads f„l, caused by the local nonlinearities . Because these are undamped one-sided springs,f„, = f„t(z) . The reduced nonlinear system equations then become :

(2.6) Mqq' + Bqq + Kqq = f9 - fni(z)

with

(2.7)

(2.8)

a t ttjfnl = [fnz(Z) OR

t~k1

fq = Lfz oR QkJ t

A good approximation will be achieved if additional eigenmodes (above Q have a neglectableinfluence on the frequency spectrum of f = fq - f„i .

The relation between the degrees of freedom x and q, respectively before and after reduction,has been given by (2 .3) : x(n., ) = T(n, x nq) q(nq) . The terms in (2 .6) have the followingdimensions : The matrices Mq, Bq and Kq are nq x nq, the columns q, fq and f,,, have lengthnq .

2.2 Some nonlinear phenomenaIn nonlinear dynamics the period Tp of some response does not have to be equal to the periodTe,, = 1/fex of the external excitation . A systems' response does not have to be periodic evenif this system is excitated with a period Tex (Thompson & Stewart ('86)[11], Hilborn ('94)[5]et . al .) .

2 .2 .1 Fundamental steady-state responses

Considering the steady-state behaviour of a nonlinear dynamical system, periodically exci-tated with TeX . Basically, one can distinguish the following responses :

. Harmonic' : The period of the response is equal to the period of the periodic exci-tation :

TP = T ex

. Subharmonic : The response repeats itself after m E N excitation periods :

Tp = mTeX7 m E N

The expressions "a 1/m subharmonic", "a subharmonic of order m", or "a period msubharmonic" all mean the same .

1 . A harmonic does, in this paper, not have to be a pure (co)sinus .

2.3 Numerical approach 5

. Quasi-Periodic : In general, this response is dominated by a (sub)harmonic com-ponent (--> fp) . But there exists at least one harmonic component, with a so-calledfree frequency ff, which is not rationally related to the dominant component with fp :ff/fp E R \ Q . For the period of the response holds :

Tp=oo

. Chaotic : The response on a periodic excitation is not periodic. The signal contains inprinciple, among the excitation frequency component, an infinite number of frequencycomponents. Therefore for the period of a chaotic response holds :

Tp=oo

Furthermore, superharmonic resonances may occur . This is a phenomenon in which oneor more higher harmonics cause resonance in a (sub)harmonic response . This is not to beconfused with superharmonic response . In a non-autonomous system, such as the systemunder consideration, superharmonic responses are not met .

2 .2 .2 Bifurcations

A variation in a design variable, for example the excitation frequency feX7 can lead to afundamental change of the response . Such changes are called bifurcations :

On the branches of periodic solutions three types of local bifurcations can be found :

. Flip bifurcation, also known as period doubling or subharmonic bifurcation .At the flip bifurcation point, a periodic solution loses stability to a solution withdouble period .

. Cyclic fold bifurcation, also known as a turning point. This is a discontinuousbifurcation . At this bifurcation point a periodic solution loses stability, and jumps toa remote periodic solution .

. Neimark bifurcation, also known as Hopf bifurcation of periodic orbits and assecondary or generalized Hopf bifurcation .The Neimark bifurcation stands for bifurcation into a torus . The system generates aso-called free frequency ff . In three dimensional state space this can be pictured astrajectories spiral around on the surface of a torus, surrounding the unstable periodicsolution . ff is measured along a cross-section of the torus, the other frequency fPdescribes the component of motion along the axis within the torus. If the ratio ff/fpis irrational, trajectories fill the entire surface of the torus : Quasi-periodic behaviour .

For this report the remark that these bifurcation mechanisms also work visa-versa will suffice .

2 .3 Numerical approach

Numerical analysis tools are indispensable for analyzing nonlinear dynamic systems .

In this report, periodic solutions are calculated by solving a two-point boundary value problemby applying a finite difference method. How the periodic solution is influenced by a changein the excitation frequency is investigated by applying a path following technique . The localstability of the periodic solution is investigated using Floquet theory .

6 Theory

User-defined settings control the solution procedures for this : convergence norms, stepsizecontrols, maximum number of iterations and the number of time discretization points deter-mine the quality of a solution . In the discussion of the results on base of nonlinear analyses,the use of symbols for such settings is necessary to make these discussions to the point .

This section provides a context for these symbols, and gives some practical insight in thesolution procedures and stability analysis relevant for this report . For an extensive descriptionof solution procedures, see Fey ('92)[3] and Parker and Chua ('89)[8] .

2 .3 .1 Periodic solutions

Periodic solutions are calculated by solving a two-point boundary value problem . Defining the2nQ column s= [qt(t) qt(t)]t representing the state of the system at time t, this boundaryvalue problem becomes :

(2.9) MQ4 + Be4 + Kqq = fq- ,,,(z)

(2.10) s(t) = s(t + Tp)

Equation (2 .9) is the same as (2 .6), (2.10) demands periodicity.

Periodic solutions are approximated using a finite difference scheme . Therefore the time t isreplaced by the dimensionless time T = t/Tp . Then T is discretized by nT equidistant pointsTi = i/nT in one period Tp .

Denote dT as a prime ( '). Then (2 .9) can be written as :

(2.11) 9(q~ q~ q , t) = 9(q" /Tp , 4/Tp, q, TpT) = 9(q~~, q 1, q~~ T) = 0

Introduce the abbreviation Q(Ti) := Qi . With a central difference scheme, 0(O-T'), the follow-ing approximations qi and q'•' for q' and q•' are used :

i qi+l - qi-1qi = 20T

-(2.12) 4", = qi+~ `Z~li + qi-~

A72

Substitution of (2 .12) in (2 .11) and using periodicity leads to the time-independent equivalentof (2 .9) and (2 .10) :

(2.13) gi=~ for i=0,1, . . .,nT-1Qi = Qi~ n,

This can be reduced by introduction of a n9nT-column z containing the nq-columns qi, for allnT discretized time points Ti :

t t(2.14) z = [4t0, . .

. This z represents the discretized periodic solution . After elimination of the contraints, (2 .13)can be reduced to :

(2.15) h(z) = 0

Thus, to determine the discretized periodic solution z, a set of nqnT nonlinear algebraicequations , given by (2 .15), has to be solved simultaneously : starting with an initial estimatezo, z can be solved from (2 .15) using a damped Newton iteration process . This process usesthe Jacobian áh/áz, size n9nT x nQn7 .

2 .3 Numerical approach 7

2.3.2 Numerical integration

If no periodic solution exists, numerical integration is used to investigate the dynamic be-haviour .

In case a periodic solution does exist, it is possible that the two-point boundary value problemconverges very slowly, because the provided start solution z° is too remote . In many cases,using numerical integration is benificial in order to find a better start solution . The Adamsmethod has been used, a variable order, variable step method (Numerical Recipes ('86)[9],Parker and Chua ('89)[8]) .

The character of a steady-state solution, calculated by means of numerical integration can beperiodic, quasi-periodic and chaotic. To quantify this character, Lyapunov exponents can becalculated . But given the large number of nonlinear dof, the calculation of Lyapunov exponentsis very cpu-time consuming . Instead, the character of a solution has been investigated bycalculating trajectories, followed by examining the patterns in the Poincaré sections as arepresentation of these trajectories . Additionally, Fourier transformations have been carriedout on some typical steady-states to quantify their characters .

2 .3.3 Definition of a Poincaré section

The Poincaré sections, used in this report are defined by :

(2.16) pk (8):= y'to+kTe: (S , t0) , k = 0, 1 . . . .

with 0 the trajectory of the state in time. Starting from an initial state s at t=t°, the stateafter k excitation periods TeX will be given by Pk(s) .

The following shorthand notation is introduced :

(2.17) Pkb (s) :_ OkTex (s, 0) , k = kb, . . . , ke , k E N

Example: " P óó °(0) for ifcnod 1 " means that the Poincaré section contains 2001 states : Fromthe state at t=500TeX to the state at t=2500Te,, . The extension ifcnod 1 means that only the1s' and the (nq + 1)th element of s are considered : z1 and zl .

The expression between brackets is the initial state, numerical integration always starts att=0 from an initial state s . In the example all initial displacements and velocities are zero .

2 .3.4 Path following

By applying a path-following (pf) technique (Fey ('92)[3]), the influence of a change in adesign variable r on the periodic solution can be investigated . This amounts to calculatingsolutions of a modified (2 .15) :

(2 .18) h(z, r) = 0

This technique uses a predictor-corrector mechanism . Starting from the "point" (Zs,k, r,,k),the kth pf-step is chosen on the tangent to the solution branch at that point, with a well chosenstepsize Qk . Substitution of the prediction (zp,k, rr,k) in (2.15) will, in general, not satisfy auser-defined convergence criterion c,, . In this case the prediction is iteratively corrected .

8 Theory

To prevent the pf-process from returning to a part of the solution curve already passedthrough, the angle ,Ql between the scaled tangent in pf-step k and the imaginary line con-necting the solution of pf-step k and pf-step k + 1, may not exceed a user-defined angle

Olmax•

Angle 02 is a measure for the difference in shape of two successive solutions . With ;Q2 < Q2maxthe pf-process can be prevented form accidentally jumping over to another branch .

The pf-process stops if :1. Stepsize smaller than minimal stepsize : Qk <6min2. The iterative correction process does not converge3. Jacobian becomes singular

2 .4 Local stability and local bifurcations

The local stability of periodic solutions is investigated by calculating the so-called monodromymatrix (Fey ('92)[3], Seydel ('88)[10]), which describes the progress of pertubations E on thestate s (s defined in (2 .10)) after one period Tp . Introduce a state s*, which lies on a periodicsolution. Denote the trajectory, starting from s* as :

(2.19) 0(t, 3*)

A trajectory starting from the perturbed state s* + So progresses with distance

(2.20) b(t) = 0(t, s* + fio) - 0(t, s*)

The distance after one period S(t + Tp) := S(T) can be approximated by means of a firstorder Taylor expansion :

(2.21) b(T) :_OO

(~s s*) bo

The matrix between b(T) and So determines whether a pertubation So decays or increases,and is called the monodromy matrix :

(2.22) 11~ (T) = aO(T, s*)as

The long term behaviour of a So depends on the 2nq eigenvalues µ; ( 1p; l > Jµ;+1 J ) of themonodromy matrix. These eigenvalues are known as Floquet multipliers. Because they cannot be zero, the progress of So after one period can be denoted as :

(2.23) b(T) = N- bo

with µ a 2n9 x 2n9 diagonal matrix with µi .

A bifurcation can be identified by means of the Floquet multipliers . At a bifurcation point,at least one of the 2nq Floquet multipliers crosses the unity circle : Iµ11 = 1. The followingbifurcations are identified by the place the crossing takes place :

Flip : l1l = -1Cyclic Fold : ml = +1Neimark : S(91) 0 0, 11111 = 1 µ2 1 , 91 = P2

2.4 Local stability and local bifurcations 9

The local stability conditions for a periodic solution are :

stable: marginally stable

unstable

In literature, characteristic exponents often appear in discussions on stability . The relationbetween the characteristic exponents ri, and the Floquet multipliers µj is given by :

= en.Tpp_

10 Theory

Chapter 3

System modelling

3 .1 Introduction

In this chapter the constitution of the model is discussed . After the model assumptions aremade, the finite element modelling of the linear part of the system, i .e. the whole systemwithout the snubbers, will be discussed in section 3 .3. Eigenvalue analyses are carried out thejudge the quality of this linear model, and to find out whether meshrefinement is required .

Next, reduction of the number of dof of the linear model is discussed in section 3 .4. Couplingof the snubbers to the reduced linear model gives us the reduced nonlinear model . This isdiscussed in section 3 .5, together with the specification of the excitation .

Appendix K contains the numerical data of the model, such as determination of stiffness,thickness, mass etc . of certain components .

3 .2 Model assumptions

An idealised solar array configuration will be examined ;

. The system is excitated by the spacecraft . This monophase base excitation is perpen-dicular to the surface of the panels .

. Because the system and the excitation are symmetric with respect to the x-z plane,only one half of the system is modelled .

. Snubbers are modelled as massless elastic elements, so no impact phenomena occur .

. There is no friction in the joints .

. Material damping is modelled by a modal damping of ~= 0 .03, based on experimentaltests performed by Fokker Space .

3 .3 Finite element modelling of the linear part

Figure 3.1 shows the general lay-out of the model. The various parts are named in table 3 .1 .

Note that one can distinguish four primary z-levels in figure 3 .1, that is a, b, c and d, respec-tively at z=-33, 0, 48, and 82 mm .

The application of a lightweight sandwich structure for the solar panels leads to a very lightstructure in combination with high stiffness . With an estimate for the representative stiffness

12 System modelling

z

d5 ----d7 ;_._...

° d á........ ... . . .tli -^ `~`~------ ------

YY

X

Figure 3 .1: Schematic view of the model

of such a composed material, a panel can be modelled with one layer of plate bending elements .The Q20SF element is used : a four-node quadrilateral isoparametric flat shell element (20 dof) .An advantage is that transverse shear stress is taken into account (DIANA ('96)[2]) .

So-called edgemembers distribute the stress smoothly at the point where a joint is connectedto a panel. Moreover they add stiffness to the solar panels . The edgemembers are modelledwith two-node 3D Thimoshenko beams (L12BE, 12 dof, see [2]), and have no mass . Allcomponents, except the panels, are modelled with these beams .

A ball and socket joint-working at b c has been accomplished as follows : b and c are verynearby points, b belongs to beam b b3, c to beam c c6 . Demanding that only the translationsof point b are one-to-one coupled to the translations of point c gives a ball-and-socket-jointworking' . The same is done for the joint at c d .

Element a1 a2 has no fysical meaning, it is only nessesary for the future reduction process .

Table 3.2 shows contraints of some important points, to make clear which type of movementthe model is restricted to .

- d3•---•d8 . . . . . ..-•-

-----•----•. .. _

1. In DIANA jargon one says :` The displacements of c are tied to b.' The command for this purpose is

"TYINGS" .

3 .3 Finite element modelling of the linear part

part "vertices"base al-a3,b1HD 1 a2, c7, d7HD 2 a3, c7, c8yoke bl-b3panel 1 c1-c8panel 2 dl-d8edgemember 1 cl, c6, c5edgemember 2 c2-c4edgemember 3 d2-d4joint yoke - panel 1 bcjoint panel 1 - panel 2 cdjoint yoke - base bl

Table 3 .1: Structure of the model. Note : a, b, c and d are distinct z-levels !

description "vertices" translations rotationssymmetry edges c4 c5, d4 d5 uy = 0 0part of base al-a3 u" uy = 0, uz = f(t) _ 0y = 0z = 0joint yoke - base bl ux, uy = 0, u,z = f(t) 0connection Hn-panel c7,d7,c8,d8 u" uy = 0

Table 3 .2: Contraints at important points. Note : u=u(x, y, z)

13

Some figures that are relevant to mention :. The panels have a thickness of 22 .36 . 10-3 m, the distance d between them is (82 -

48- 22 236 x 2) • 10-3 = 11.64 . 10"3 m .. The diameter of the yoke is unknown, therefore the distances base-yoke and yoke-panel

1 are unknown .. The total mass of the The total mass of the finite element model is 11 .02 kg, which is

half the mass of the real system because only one half is modelled .

3.3.1 Eigenvalue analysis of the linear model

At this stage, an eigenvalue analysis of the linear model has been carried out in order to judgeits quality and to find out whether meshrefinement is required .

Earlier studies, done by Fokker Space b .v. indicated that the first eigenfrequency of the systemwithout snubbers had to be in the range 30 - 40 Hz .

First, an eigenvalue analysis has been performed on a coarse, basic 3D FEM-model (1692 dof) .In order to get some idea of the accuracy of these eigenfrequencies, meshrefinement has beencarried out . This led to a second model with 5894 dof.

To give an idea of the element distributions of these models : for the 1692 dof model eachpanel consists of 16 x 8 = 128 elements, and 10 elements for the yoke . For the 5894 dof modelthis is respectively 32 x 16 = 512 and 20 elements .

One can expect that especially the higher eigenfrequencies calculated for the 5894 dof modelwill differ from the basic model .'Table B .1 displays the eigenfrequencies of both models . The

14 System modelling

eigenfrequencies [Hz]no . 1692 dof 5894 dof

1 32.751 32.1612 42 .648 42.6333 75 .487 75.5524 105.47 99.3275 129.82 123 .636 140.07 132 .357 144.53 135 .398 154.93 157.439 170.02 164 .47

10 173.68 167 .5611 188.56 186.2212 209.66 203 .5513 213.80 207.5414 245.79 244.7315 246 .06 245.2516 269 .96 275.35

Table 3 .3: Eigenfrequencies of the base-supported linear model before and after meshrefine-

ment

eigenfrequencies within the frequency range of interest (20 - 100 Hz) do match quite well : nofundamental other eigenfrequencies show up after the meshrefinement, so the second columnof table B.1 is the accurate version of the first column .

Further meshrefinement does not pay off, certainly not for the eigenfrequencies within therange 20-100Hz . In dialogue with Fokker, it had been decided to do no further meshrefinementand to accept the 5894 dof model for further analysis .

3 .4 Reduction of the linear model

The number of degrees of freedom of the linear model will be reduced by application ofComponent Mode Synthesis . The pseudo-load method will be used . The mathematical outlineof this process was sketched in section 2 .1 .

Here, the discussion is restricted to the practical approach :

The whole linear model has been taken to be one unsupported (sub)structure (eigenfrequenciesshown in table 3.4) . As mentioned, the real system contains 8 snubbers . Because of symmetry,the model representing only one half of the system must be prepared to contain 4(!) snubbers,mounted between 2 x 4 = 8 nodes of the model (fig . 3.2) . These so-called interface nodes(ifcnod(s)) alone are responsible for 8 residual flexibility modes since only the dof in z-directionare relevant . Hence, the component exhibits one rigid body mode in z-direction. The numberof dof of the reduced component nq depends on the number of modes included in the dynamic

3.4 Reduction of the linear model 15

no. f„ [Hz] no . f„ [Hz] no . f„ [Hz]1 0.0003 6 123.64 11 167.592 37 .504 7 132.68 12 186 .243 42 .734 8 139.62 13 204.014 75 .627 9 157.56 14 240 .645 103.43 10 164.47 15 244.79

Table 3 .4: Eigenfrequencies of the free linear component

component mode set, defined by (2 .2) . This justifies the following equation :

(3.1) nq = nboundary dof + nrigid body modes + nkept free-interface eigenmodes

= nB+nR+nk =8+1+nk

= 9 + nk

In this case, nq is fully determined by nk which depends on the cut-off frequency fc . As statedearlier, the frequency range of interest is 20-100 Hz . The choice of f, is a compromise betweenlong cpu times and accuracy of the solution in future nonlinear analysis . In general, fc will bechosen higher than the highest excitation frequency to avoid resonances below fc, caused byartificial eigenfrequencies corresponding with the residual flexibility modes .

Table 3 .5 shows the performance of the reduced linear model, consisting of the reduced linearcomponent connected to a rigid base, compared with the original linear model (5894 dof) forthree values of fc. In this configuration all base nodes are fully supported to suppress the rigidbody mode, which in future analyses also will be absent because the system is subject to aprescribed base displacement derived from the base acceleration 2 .

Counting the number of eigenfrequencies up to f, in table 3 .4, added with the inevitablenB = 8 results in the total number of dof mentioned in table 3 .5 .

Table 3 .6 shows which points of the original model (fig . 3 .1) correspond to the ifcnods of thereduced model (fig . 3 .2) .

Some details on adjustments to the model, needed for maximum reduction

To reduce the number of ifcnods to the absolute minimum of 8, a special adjustment to themodel is nessesary.

The base of the unreduced model consists of the points al (lower connection of snubber 1),a2, a3 and bl (see figure 3 .1) . If after reduction from the 'base nodes' only al (-> ifcnod 0) issaved, external excitation of ifcnod 0 has to be equivalent to in phase base excitation of thejust mentioned base nodes . This is achieved by tying the z-displacements of a2, a3 and blto the (prescribed) z-dof of al . Saving al as single base node is very benificial because nowtwo usefull purposes are combined by this interface node : interface for external excitation andinterface for connecting a snubber to the reduced linear model

Further, a stiff massless beam is placed between a2 and al . This is nessesary for reduction ofthe whole linear model at once .

2. In principle, this is not true for feX = 0 Hz . Integrating the periodic base acceleration with constantamplitude two times gives an expression for the prescibed displacement that heads to infinity (i.e. rigid bodymode) for excitation frequency zero .

16 System modelling

no . Original5890 dof

fr = 200 Hz20 dof

Rel.err[%]

f, = 125 Hz14 dof

Rel.err[%]

f, = 100 Hz13 dof

Rel.err[%]

1 32.161 32.161 + 0 .000 32.161 +0.000 32.161 +0 .0002 42.633 42 .633 +0.000 42.633 +0.000 42.633 +0 .0003 75.552 75 .552 +0 .000 75.552 +0.000 75.552 +0 .0004 99.327 99 .328 +0 .001 99.361 +0.034 99.362 +0 .0355 123 .63 123.63 +0.000 123.64 +0.008 130.84 +5.8316 132.35 132.35 +0 .000 132.51 +0.120 157. 74 +19.187 135.39 135.40 +0 .007 156.80 +15.81 172.78 +27.618 157.43 157.43 +0 .000 188 . 4 1 +19.67 188.66 +19.839 164.47 164.47 +0 .000 194.27 +18.11 197.62 +20.15

10 167.56 167.56 +0.000 199.64 +19.14 201.60 +20.3111 186.22 186.22 +0.000 224.09 +20.33 226.21 +21 . 4 712 203.55 203.73 +0.088 293.70 +44.28 484.90 +138.213 207.54 210.92 +1 .628 447.11 +115.514 244.73 313.25 +27.99 494.43 +102.015 1 1 245.25 316.33 +28.98

Table 3 .5 : Comparison of the reduced and unreduced system . The component modes of thefree (unsupported) component, corresponding to the eigenfrequencies summarized in table3.4, are used to describe the displacement field of the reduced linear model with supportedbase. Up to the cut-off frequency, the eigenfrequencies of the reduced linear model and theunreduced linear model are almost equal (eigenfrequencies higher than f , in italic) .

Excitation of ifcnod 0 of the reduced model is now equivalent to in phase excitation of all thebase-nodes of the original model .

3 .5 Completing the nonlinear model

This section describes the implementation of the local nonlinearities and the external excita-tion in the reduced linear system equations . This means respectively filling in the right-handside of (2 .6) which will make the reduced system equations nonlinear, and application of aprescribed displacement(!) to ifcnod 0 .

Original (fig.3.1) Reduced (fig.3.2)

~ snubber 1

~ snubber 2

~ snubber 3

~ snubber 4

Table 3 .6: Transition to reduced model

3.5 Completing the nonlinear model 17

3.5.1 Modelling of the snubbers

The nB-column z containing the interface dof of the reduced model, see (2 .5), will be definedby :

(3.2) z = [ZOI Z11 . . . , z7]t

A user-subroutine NDSPTR .F (appendix L.2, page 83) has been written to describe the one-sided springs :

fsnub =~0 if x< d,ksnub x+ bsnub x if x> d .

with x = zj - zj_l, j E{1, 3, 5, 7} (see fig .3.2) and d the backlash . Notice that the disconti-nuity in stiffness is not approached by some arc-tangent . Calling f$nubj the snubber-force onifcnod j, the nq-columns fml and fq , see (2 .7) and (2 .8), become :

fsnub,0

fsnub, t

fsnub, 2

fsnub,3

fsnub,4

(3 .4) fnl(Z) = fsnub,5 , fq = 0fsnub,6

fsnub,7

0

0

The 18t and 2nd element of f„ , are opposite, as are the 3rd and 4th, the 5`h and 6th, etc. Thecolumn fq contains only zeros, because the system is subject to a prescibed base acceleration .

3.5 .2 Excitation

The system is excitated with a prescribed harmonic base acceleration with an amplitude ofA= 2g := 20 ms-2 . The following relations, which hold for ifcnod 0, have been implemented

in the user-subroutine USRLOD .F (appendix L .1, page 81) :

-(3.5) zo

_

-1

(2rf )2 A cos(2r T + (p)ex

(3.6) zo = 2 f A sin(2rT + W)ex

(3.7) zo = A cos(2rT + w)

(3 .10)

Ozoafexaioafex020afex

12~2 f3 A cos(2~rT -f + - W)

ex

2rf X A sin(27rT + co)

0

18 System modelling

For every calculation holds: cp = 0 . The dimensionless time f = fe,,t (54 -r = fpt) .

The first deratives of the external load to the excitation frequency, (3 .8) to (3.10), are neededto investigate how a periodic solution is influenced by a change in the excitation frequency . Itcan be seen that the displacement amplitude decreases quadratic with increasing frequency .In case feX 10, z-* oo, i .e. rigid body mode. Stiffness, damping, and backlash can be varied insubsequent analyses because the first deratives of the external force function to these designvariables are implemented too (snubber damping implies that f,bi = f„i(ti, z) . But since thisdamping will be zero in all analyses, the dependence of the snubber force on the velocity hasbeen omitted in the discussion) .

Simulations with snubbers under pre-stress can be achieved by choice of a negative backlash .

3.5 Completing the nonlinear model

z

x

~

Í

5-83-{

X-1,

7 ----------------- . . . . . .. _ - . ---- .--° ` 5

Top View

O3-52-2

Side View

2)

2

x

19

Figure 3 .2: Schematic view of the reduced model

20 System modelling

Chapter 4

Linear dynamic analysis

.

A linear dynamic analysis has been carried out, using the reduced model with 14 dof (f, =125 Hz). The four nonlinear elements have been replaced by linear two-sided translationalsprings . This model is exposed to in-phase harmonic base excitation with a constant acceler-ation amplitude of 2g .

Frequency responses were determined in three cases : k = 0, k = 5 .a104 and k = 1 .a105 Nm-i,k represents the linear stiffness of all four springs . Table B .2 contains the eigenfrequen-cies for these cases . Appendix A displays the eigenmodes of the system for k = 0 Nm-1 .The path following routine (section 2 .3 .4) has been used: n,. = 600, /31 = 02 = 30°,Ez=10-9and10-10<vk -<, l .

Because the investigated system is linear, the linear prediction of the solution at a distanceQk immediately satisfies the convergence criterion, no iterative correction is needed . Thereforethese analyses run very smooth, no hard failure of the pf process occurs .

The first case (k=0) is shown by figure 4 .1 . Figure 4.1a shows the base input at ifcnod 0,according to (3 .5), and the response of ifcnod 1 . The figure title only indicates that in futurenonlinear analyses snubber 1 will be placed between the ifcnods mentioned in the figure. Thefirst resonance peak shows up at 32 .2 Hz (= first eigenfrequency, see table B .1), the secondeigenfrequency at 42 .6 Hz is only marginally visible. This is due to the fact that in the firsteigenmode ifcnod 1 moves mainly in the x-z plane, in the second eigenmode it moves mainlyin the x-y plane and thus has no component in the z-direction . Figure A. ja shows this . FigureA.2a represents the third eigenmode, and explains the peaks at 75 .3 Hz for ifcnod 4 and 6 infigure 4 .1' and 4 .1d respectively . Interpretation of what is going on at other resonance peakscan be done in a similar way.

Consider figure 4 .1b. Within the frequency range of interest, 20 - 100 Hz, no perceptibledifferences occur between the maximum of the amplitudes of ifcnod 2 and 3 . The pictures ofthe eigenmodes show that up till the sixth eigenmode at 132 Hz these ifcnods move in phase .This all means that application of snubber 2 serves no purpose .

Remember the space between the two solar panels is d = 11 .64•10-3 m. Consider for exampleifcnod 4 and 5 (figure 4.1'-) : The panel ends strike each other if :

(4.1) Z4 - zs > d

In terms of IzImax however, whether the panel ends strike each other depends on whether theymove in-phase or out of phase :

22 Linear dynamic analysis

. If they move in-phase, they strike each other if :

(4.2) (Jz4lmax - IZSlmax) > d

. If they move out of phase, they strike each other if :

(4.3) I Z4lmax + IZ5Imax > d

The maximum difference in amplitude occurs between ifcnod 6 and 7 at 32.2 Hz, the firsteigenfrequency: in the worst case, ifcnod 6 and 7 move out of phase (hard to see in eigenmode-plot fig . A.la) . Using (4 .3), a minimum distance dmin between the panel ends can be calculatedas follows :

dmin = d - ( IZ6Imax + IZ7Imax)

= (11.64 - (6 .6 + 0.53)) . 10-3= 4.5•10"3 m

Conclusion : The solar panels of the linear model, without snubbers, exposed to in-phaseharmonic base excitation with a constant acceleration amplitude of 2g, will never strike eachother for excitation frequencies within 20 - 100 Hz .

Figure 4.2 shows two known tendencies : with increasing spring stiffness eigenfrequencies shiftto higher values, and displacement amplitudes decrease . For amplitude-frequency plots of theother ifcnods, one is referred to figure B .1 .

23

1 z 1 m,x[m] snubberl a . 1 z 1 mex[mI snubber2 b.

10°n~«+, 104

104Input at~ 0 :~ z ~,,,,_2glc4xf,a°

~

104W~3

10 4

"' na,od 2

25 50 75 100 125 fax [H Zl 25 50 75 100 125 fax [H Z]

~ z 1 ..[M] snubber 3 C . ~ z' m°x [ml snubber 4 d .%iknod4 nMrnodB .

10' 10,

6 Mcnod 7 '.,

104 104

25 50 75 100125 f®x [H Zi

25 50 75 100 125 fax [HZ]

Figure 4.1 : Frequency responses on the basis of linear dynamic analysis : k=0 Nm-i

24 Linear dynamic analysis

1 z 1 max rml ifcnod 1k--o

1 =5.0E4i^ k=1 .OE5 Nrri'

\ i i i\ i \

~.Y

I I I I -'- __ I

25 50 75 100 125 fax [H Zi

Figure 4.2 : Frequency responses for ifcnod 1, on the basis of calculations with k=0, k=5 .0•104and k=1 .0 .10' Nm-1

Chapter 5

Nonlinear dynamic Analysis

5 .1 Introduction

5 .2 Considerations on the solar array system

Implementation of a snubber causes a local nonlinearity. Consider the ratio

ksnub(5.1) a = kj

with ksnub the snubber stiffness, assuming that all snubbers are identical, and k j the stiffnessof the linear system at the spot of ifcnod j ; kj follows from the reciprocal of the displacement,caused by a unit force on ifcnod j .

This a = a(ksnub) is a measure for the degree of nonlinearity . The real snubber stiffness is

ksnub = 1•107 Nm-1 . In general however, the higher the degree of nonlinearity, the more difficult

it is to find periodic solutions . For this reason the intermediate cases with ksnub = 1 .10' and1 .105 Nm-1 are introduced .

At snubber 3, ifcnod 4 and 5, the highest relative snubber stiffness occurs, see table 5 .1 . Inan attempt to concretize values of such an a, the following is found in Fey ('92)[3] :

a = 1, Moderately nonlineara = 6, Strongly nonlinear

One can expect weakly nonlinear system behaviour with ksnub = 1•104 Nm-1 #, 0.1146 < a <0.3118 . Anticipating on the results in section 5 .4, the setup with ksnub = 1•105 Nm-1 is calledstrongly nonlinear . A snubber stiffness of ksnub = 1 .107 Nm-1 leads to 114.6 < a < 311.8 :very strongly nonlinear .

In a single degree of freedom system with only one local nonlinearity of type "one-sided linearspring", harmonic resonance occurs near the bilinear eigenfrequency

(5.2) fb =2V-1 -+a fl

1 + 1+a

if and only if

1 . backlash is zero, i .e. both snubber ends pass equilibrium simultaneously in case of oneone-sided spring .

2 . system is undamped3 . system is unforced

26 Nonlinear dynamic Analysis

ifcnod j kj [Nm-1] a(ksnub = 1• 105) [-]1 4.5517E+04 2 .1972 3 .8911E+04 2 .5703 3 .8521E+04 2 .5964 3.4650E+04 2 .8865 3.2072E+04 3 .1186 8.7260E+04 1 .1467 7.0028E+04 1 .428

Table 5 .1: Values of a . For ksnub =k , a( k ) = ó a(105)l

In that case fb can also be derived as follows :

(5.3) Total period time T=T° + Tb

2_ 1 _ fa fb

(5.4) fb _ T _ 2 fa -I- fn

with fa and fb the first eigenfrequency of the linear system, respectively with and without atwo-sided spring attached .

Consider snubber 4: it is not possible to calculate fb for ifcnod 6 or 7 because the presenceof the other snubbers make it impossible to separate a linear case a and b as above . Onecannot expect that all snubbers contact at the same time . If for example snubber 4 contactslater than snubber 2, the internal forces, generated by the force-discontinuity at this snubber,distribute and feel like external forces at snubber 4 .

This all means that the solar array system does not fulfil criterion 1 and 3 . This is thereason one cannot speak of the eigenfrequency of this system, because systems like these aresubject to their own "internal excitation"' : instead one speaks of harmonic resonance, whichbelongs to the wide variety of behaviour a nonlinear dynamic system can exhibit . The bilineareigenfrequency will only provide a rough estimate of the first harmonic resonance frequency .

Further, the snubber force has been defined in such a way that it is non- differentiable forx = d, see (3.3). Because of this discontinue formulation, the consistency of the approximationof the periodic solution z is 0(Dr2) . One cannot cheaply improve the accuracy to 0( LT4) bymeans of a deferred correction technique ( Fey ('92)[3]) : for estimating the local discretizationerror E9i one needs an estimation for the third and fourth time derative of the displacement,respectively q;3~ and q;4~ . At the discontinuity in stiffness, the derative of q(t) (N q;" ) isundefined .

5 .3 Weakly nonlinear

This section handles the case ksn„b = 1•104 Nm-i, 14 dof (f,=125 Hz) and 0 .1146 < a< 0 .3118 .

Numerical solution parameters : nT = 600, ~31 =~32 = 30°, c ;, = 10-9 and 10-9 < Qk <, 2 .

1 . The usage of the term internal excitation here is not correct . In this report however, it will always be usedin the way it is used here .

5.4 Strongly nonlinear 27

Consider figure 5 .1 . Nonlinear dynamic analysis of the system with the considered configura-tion demonstrates that :. All solutions are stable .. The first harmonic resonance occurs at ±34.2 Hz. (32 .2 Hz in case "linear, no spring"). Despite the weakly nonlinear nature of the system, superharmonic resonance peaks

can be distinguised : The left inset of figure 5.1 shows a 2°d superharmonic resonance,originating from the 1St harmonic resonance peak . The right inset shows a superhar-monic from an artificial eigenfrequency (above Q originating from a residual flexibilitymode .The occurance of this resonance has no physical relevancy, because it is caused by anartificial mode, applied to improve the accuracy in the frequency domain below f, .

. The response curves of ifcnod 2 and 3 are almost identical in the frequency rangeof interest : snubber 2 has little functionality. Near 34 .2 Hz some marginal differenceoccurs between them . However, the superharmonic resonance "spike" at 70.7 Hz seemsto be damped by snubber 2 .

5 .4 Strongly nonlinear

5 .4.1 Introduction

This section handles the case kg,,,,b = 1•105 Nm-1, 14 dof (f,=125 Hz), and 1 .146 < a< 3 .118 .

The frequency responses of ifcnod 1 and 6 in terms of the maximum of the absolute z-displacement, see figure 5 .2, will be used to illustrate the kind of dynamic behaviour one canexpect at its various branches . For the amplitude-frequency plots of ifcnod 2, 3, 4, 5, and 7,see figures C .1-C .3

In section 5 .4 .2, the frequency response of the system is globally discussed using the amplitude-frequency plot of ifcnod 1 . For explanations and a detailed discussion of the phenomena thatare found, one is referred to section 5 .4.3 .

5 .4.2 Global discussion of system dynamics

The frequency response of ifcnod 1(fig . 5.2a) will be used to globally discuss the dynamics ofthe system under consideration . The o-symbols represent unstable solutions .

Ifcnod 1 is part of the yoke . "Excluding" the influence of the other snubbers elsewhere inthe construction, the "yoke-snubber 1"-system is very similar to the "beam supported bya one-sided spring", discussed by Fey ('92)[3] . This explains the somewhat familiar look offigure 5 .2a .

Considering fig . 5.2a, a second superharmonic resonance peak shows up at 21 .9 Hz, caused bythe first harmonic resonance peak a 43 .8 Hz. At ±33 Hz, a stable 1/2-subharmonic solutionbranch can be expected, induced by the mentioned superharmonic resonance at 21 .9 Hz. Thisbranch has not been found . Instead, a stable 1/3 subharmonic solution between at 34.75 Hzhas been found. In terms of amplitude, this solution can hardly be distinguised from theunstable harmonic solution in this frequency range.

For the other unstable regions on the harmonic solution branch up to fe7C = 74 .94Hz holds : thestable steady-state behaviour found within these regions is very similar to the accompanyingunstable harmonic solutions .

28 Nonlineardynamic Analysis

Within the range 74 .94- 98 .2611z , 1/2 subharmonic solutions have been found, which bifurcatefrom the harmonic branch via flip bifurcations at the boundaries of this interval . The harmonicbranch is unstable in this range. The ratio of the amplitudes on the 1/2 subharmonic and theharmonic branch has a maximum at feX = 87 .7 Hz :

I Z1lmax,1/2 sbh - 6•5•10'4

- 9.3

I Z1 I max,harm 7•0' 10-5 ~

Between 91 .78 - 97.93 Hz, a 1/4 subharmonic solution branch exists, bounded by two flipbifurcation points on the 1/2 subharmonic branch. Qua amplitude, this branch differs justa little the 1/2 subharmonic, but interesting phenomena, such as Neimark bifurcation of astable 1/4 subharmonic solution in a quasi-periodic "torus" have been encountered .

At feX = 99.13 Hz, the response turns out to be chaotic .

Again holds: snubber 2 has litte functionality, see figure C .1 .

Furthermore, perceptable superharmonic resonances occur at 49 .7, 62.0, 68.0, 82.4, 99.1 and123 Hz

5 .4.3 Detailed discussion of the encountered phenomena

Systems with several local nonlinearities of type one-sided spring can be "internally excitated",in the way this is mentioned in section 5 .2. For a demonstration of this effect, one is referredto appendix D. The harmonic solution at 20 .48 Hz has been used for this purpose .

Consider the harmonic solution branch for ifcnod 1, see fig . 5.2. This branch is unstable inthe range feX = 34 .60 - 34 .63 Hz, the edges of this interval are Neimark bifurcation points . At34.61 Hz, "maximum instability" is reached : lµlI = 10 .394 f 1 .08i) = 1 .15. For fex = 34 .64 Hza stable harmonic solution exists .

Flip bifurcations mark the unstable region 34 .65 - 34.69 Hz, lµl I = 1 .49 at 34 .65 Hz .

The stable region 34.69-34.71Hz is followed by the unstable region 34 .72-34.76Hz, marked byflip bifurcation points . Numerical integration at feX = 34 .75 Hz provided a start solution thatled to a stable 1/3 subharmonic solution . Since flip bifurcations can not lead to this solution,it is obviously that cyclic fold bifurcation points exist on the 1/3 subharmonic solution branchexist just before and after 34.75 Hz . The explanation of the invisibility of this branch in e .g .figure 5 .2a. is given by figure 5 .4 : The unstable harmonic and the stable 1/3 subharmonicsolution are almost equal . The closing up unstable region, from 34.77 - 34 .83 Hz, is initiatedby a Neimark bifurcation . One can expect quasi periodic-behaviour . Numerical integrationat feX = 34 .8270 Hz, Adams method, 9 significant digits, so = [0 0]t for 1000Tex, resulted inthe Poincaré section of figure E .1: It looks like quasi-periodic behaviour has been locked on a1/17 subharmonic . Figure E .3 shows the Floquet multipliers for the just discussed frequencyrange .

Not for all unstable ranges on the harmonic solution branch the stable steady-state behaviourhas been investigated. In terms of amplitude, the stable steady-state behaviour does not differsignificantly from the unstable harmonic solution within the range 34 .83 - 74.93 Hz .

Within the frequency interval 74 .94 - 98.26 Hz no stable harmonic solutions exist . A 1/2subharmonic solution branch has been found, surrounded by flip bifurcation points on the

5 .4 Strongly nonlinear 29

harmonic branch . At fe1 , = 87.7 Hz, the 1/2 subharmonic solution has maximum amplitude :

Izll ma,, = 6.5•10-4 m. See figure E .4 for some typical phase portraits in the neighbourhood ofthis peak .

Again not all unstable regions on the 1/2 subharmonic branch have been investigated . Butthe unstable region 95.51 - 97.93 Hz is marked by two flip bifurcation points on the 1/2subharmonic branch (fig. F.6) . In this interval a 1/4 subharmonic solution exists . This branchis better visible in the amplitude-frequency plot of ifcnod 6 (figure 5.2b) . Zooming in on theboxed area gives figure 5 .3. For some typical stable 1/4 subharmonic solutions, see figure F .1 .The unstable region 96 .808 - 97.018 Hz on 1/4 subharmonic branch is marked by Neimarkbifurcation points, see figure F .3: at feX = 96.808 Hz Neimark bifurcation of a stable 1/4subharmonic solution into a quasi-periodic "torus" is found . This quasi-periodic solutionshows four closed curves in the Poincaré section, see figure 5.5 and figures F.4 - F.5. Interms of amplitude, the 1/4 subharmonic branch can hardly be distinguised from the 1/2subharmonic solution branch. This is especially true in the neighbourhood of the Neimarkbifurcation points, see figure F.2 .

At feX = 99 .136 Hz, figure 5 .3 displays a sharp peak. In the direct vicinity of this peak, nostable periodic solutions exist . Examination of the Floquet multipliers, corresponding withthe unstable harmonic solution for fe,, = 99.136 Hz shows that one of the Floquet multipliersreaches the extreme value I µl I = I + 45 .8 ± 0.00i 1 . The stable steady-state behaviour for thisexcitation frequency has been investigated by means of time integration, and turns out tobe chaotic . Figure F .7 depicts the Poincaré sections for this excitation frequency, figure F .8displays some time histories and phase portraits of the chaotic 'solution' . In practice, thechaotic nature of the response will be noticed by an irregular in- and decrease of amplitude intime, see fig . F.8a. The way, in which transition to chaos takes place, has not been investigated .Neither are the boundaries of the frequency interval in which chaos has been encounteredexactly known. Estimates of the upper and lower boundary of the frequency interval in whichchaos exists are 98 .4432 and 102 .922 Hz respectively . These estimates are based on the Floquetmultipliers of the harmonic solution branch (table 5 .2) .

For excitation frequencies exceeding the frequency range of interest, some unstable regions onthe harmonic solution branch have been found : The intervals 110 .7 - 116 .9 Hz and 123 .7 -126.8 Hz are bounded by flip bifurcation points . The unstable region 121 .5 -123 .1 Hz is markedby Neimark bifurcation points . For these intervals, no subharmonic solution branches havebeen found .

Due to the resemblance of the frequency response of ifcnod 1 and the amplitude frequencyplots found for the "beam supported by a one-sided spring"-system discussed by Fey ('92)[3],one can expect a so-called 1/3 subharmonic solution island near three times the lst harmonicresonance frequency ( -- 130 Hz) .

In order to find this eventual stable 1/3 subharmonic solution, several attempts have beenmade: using numerical integration starting from a state of an unstable harmonic solution andstarting with initial displacements and velocities zero never led to "period three behaviour" .

The frequency responses each contain over 3100 branch points, calculated in 46 "path followingruns" . In order to get some idea of the effort invested in these frequency responses, one isreferred to appendix I .

30 Nonlineardynamic Analysis

range µl stability98 .2633 - 98 .4432 jµl I < 1 stable98 .4432 - 98 .5032 11 cl I > 1, Q01) # 0 unstable98 .5023 - 102.922 R(µl) < -1, s(µl) = 0 unstable

Table 5.2: Stability of the harmonic solution branch in the neighbourhood of the chaoticbehaviour .

5 .4.4 Fourier transformations

The frequency spectrum of the response at feJC = 99 .136 Hz has been determined by meansof Fast Fourier Transformation, in order to 'prove' the chaotic nature of the response . Asmentioned, calculation of Lyapunov exponents would be too expensive in terms of cpu time .

Comparison of the frequency spectrum of a harmonic solution with the spectrum of the chaoticsolution, see figure 5 .6, clearly shows that in case of chaos every frequency has a significantcomponent in the frequency spectrum of the response .

For more frequency spectra, one is referred to appendix G .

5 .5 Very strongly nonlinear

This section handles the case kgn„b = 1•107 Nm-1, 14 dof (fc=125 Hz), and 114.6 < a < 311 .8 .

This value for the snubber stiffness is determined experimentally by Fokker . Given the a's,one can expect very nonlinear dynamic system behaviour . It appeared to be very hard tofind periodic solutions for the system under consideration . The main reason is the very shortcontact time which will be defined as the time span in which a snubber is compressed :For application of the finite difference method, the dimensionless time r = t/TP is divided inn,, equidistant time points ra (section 2 .3.1). The most abrupt state-changes occur if a panelcontacts a snubber. The contact time decreases with increasing snubber stiffness . For thisreason one has to increase nT dramatically in order to have enough time discretization pointsin the contact time interval to describe the sudden state changes . Increasing n,, leads to amore than proportional increase in cpu time . For this reason it was not practicable to workthis case .

The assumption is made that no impact phenomena occur . For very high snubber stiffnessthe impact effect of a real snubber can become dominant to the stiffness effect . Due to thementioned assumption acceleration levels exceed 200 ms-2 for a model with ksnub = 105Nm-'(fig. F.1-f) . For simulations with ksnub = 107 Nm-1 it should be wise to take impactinto account .

31

Figure 5.1 : Frequency response (14 dof), ksnub = 1 .10' Nm-i (weakly nonlinear) .

32 Nonlinear dynamic Analysis

I Z I ... [m][ 2nd superh.

10"3

ifcnod 1

harmonic

'/2 subharmonic

10"4

25

I Z I ... [m]

50 75

ifcnod 6

10'3

100

'/2 subharmonicharmonic

L - 1 I

25 50 75 100

a.

125 feX [H z]

b.

125 fex [H z]

Figure 5 .2 : Frequency responses of ifcnod 1 and 6 . Unstable branch points are marked by o's

5.5 Very strongly nonlinear 33

Figure 5 .3: Enlargement of the boxed area of fig . 5 .2b

Figure 5 .4: Harmonic (dotted) and 1/3 subharmonic (solid) of ifcnod 1 and 7 at fe7C = 34 .75 Hz(n, =2400). For the harmonic solution holds : Jµ,1 _ ~- 1 .54 ± 0 .00i I = 1 .54

34 Nonlinear dynamic Analysis

Velocity [MS71]

j:~~2

-0 .025

-0 .050

-0 .075

~ . .-4.0000E-5 -3.0000E-5 -2.0000E-5

0disp . [m]

Figure 5.5: Quasi-periodic behaviour within the frequency range 96 .808-97.018 Hz . This figureis a Poincaré section P2600000 0(6) of ifcnod 5, feX = 96 .900 Hz .

I z, I [m]

104

10-1- chaotic

104

10''10-1-10-1-10-10 . harmonic

10•„ ;fww10•,2

~_40 50 100 150 200 250 300 f[HZ]

Figure 5.6 : The harmonic signal (fe%=58 .000 Hz) is determined by one frequency component.The contribution of other frequency components to the signal is < 10-1Ó -round-off error . Thechaotic signal (fe,t=99 .136 Hz) is composed of an infinite number of frequency components .Every frequency has a significant component in the frequency (at least 1000 x higher comparedto the harmonic signal) .

Chapter 6

Comparison nonlinear / linear

In this chapter the results on base of the nonlinear dynamic analyses with k9nub=105 Nm-i willbe compared to the results of the three linear dynamic analyses (k=0, 5• 104 and 105 Nm' 1) .In figure 6 .1 the corresponding frequency responses of ifcnod 1 are collected . For a completesurvey of all ifcnods, one is referred to appendix H .

Strictly speaking, there is no ground for comparing these two cases, since the nonlinear systemcan not be linearized : the nonlinear analyses do not serve as a reference for linear analyses ofa well chosen linearized case .

Here, plotting frequency responses on base of three linear cases together with the frequencyresponse of the nonlinear system serves an important purpose : it emphasises the occuranceof tremendous errors if one wants to simulate the dynamic behavior of a system with localnonlinearities by some linear "equivalent" .

In fact, one can not approximate the dynamic behaviour of the nonlinear system by replacingthe snubbers with linear 2-sided springs . Given a certain snubber stiffness, one can only ap-proximate the first harmonic resonance frequency by the bilinear eigenfrequency (see (5.2)) .Recall that this equation only holds for the bilinear case . An approximation for the lst har-monic resonance frequency of the nonlinear system with ksnub = 105 Nm-1 and consideringifcnod 1, can be obtained by :

(6.1) a= 2.197, (table 5 .1, page 26) ~ in eq. (5 .2) ~ fbl = 41 .251 Hz

fl = 32 .161 Hz, (table B .1, page 46)

In this case, the bilinear eigenfrequency is a reasonable estimate of the 1st harmonic resonancefrequency of the nonlinear system, which occurs at fe7C = 43 .8 Hz. But this approach does notprovide any information of e .g . the height of the peak, or the error of the approximation dueto illicit application of (5 .2) .

Figure 6 .1 points out that the frequency response of ifcnod 1 of the nonlinear system is closestto the linear case with k ;zz~ 3•104 Nm-1 . But since a linear system cannot exhibit subharmonicresponses, at fe7C = 88 Hz the difference in amplitude between the stable 1/2 subharmonicresponse and the response of any discussed linear linear case is at least a factor 10 .

36 Comparisonnonlinear / linear

Figure 6.1: Comparison of linear dynamic analyses (k=0, 5 . 104, 105 Nm-1) and nonlineardynamic analyses (ksnub = 105 Nm-1)

Chapter 7

Conclusions & Recommendations

7.1 General conclusions

. It can be dangerous to rely on results based on linearized systems that can not belinearized, such as the considered solar array system with snubbers . This is especiallytrue for a high degree of nonlinearity .

. DIANA appeared to be a very usefull tool for steady-state nonlinear dynamic analysesof multi-degree-of-freedom models .

7.2 Case specific conclusions

It is not practical to investigate the case with kgn„b = 107 Nm-1, which is the real snubberstiffness, using the finite difference method with equidistant time discretization to solve thetwo-point boundary value problem . One needs too much time discretization points to have atleast some in the very short time interval in which contact takes place . The shooting method isno alternative because the nonlinear model has to many degrees of freedom : several attemptsusing shooting method (simple and multiple shooting) lead to unacceptable long cpu-times .

For the strongly nonlinear case (ks,,,,b=105 Nm-1) the following conclusions can be drawn :

. Nonlinear dynamic analyses showed very rich dynamic behaviour . In practice, espe-cially the 1/2 subharmonic solution branch is important because it spans such a widefrequency range (74 .94 - 98 .26 Hz). The difference in amplitude between this solutionand the accompanying unstable harmonic solution reaches a maximum at fe1C = 87 .7Hz,where the amplitude of the 1/2 subharmonic solution exceeds the amplitude of thecorresponding harmonic solution by a factor 10 .

. Comparing the results between nonlinear and linear dynamic analyses showed that itis not possible to estimate the dynamics of the nonlinear system by replacement ofthe snubbers by standard linear springs with well chosen stiffness values .The comparison reveals that the differences in amplitude corresponding with threelinearized cases and the solutions calculated by means of nonlinear dynamic analysiscan exceed a factor 10 . This holds especially near the peak of the 1/2 subharmonicsolution branch .

. Application of snubbers mostly leads to diminished resonance and anti-resonancepeaks . A snubber especially reduces the displacements that lead to its compression .

38 Conclusions & Recommendations

. Snubber 1 is most functional, because ifcnod 1 has the highest amplitude . Snubber 2is least effective, and can be omitted because it hardly affects the calculated frequencyresponses .

. Using a prescribed base acceleration with an constant amplitude of 2g, the panels willnot strike each other if no snubbers are applied .

. Neimark bifurcation of a stable 1/4 subharmonic solution into a quasi-periodic "torus"has been found .

. Near fe7C = 99.136 Hz the system reveals chaotic behaviour .

For the weakly nonlinear case (kgnub=1O4 Nm-1) no special phenomena occured : for eachfrequency within the frequency range of interest only one stable harmonic solution exists .

7.3 Recommendations

. To work out the very strongly nonlinear case (k8n„b=107 Nm-i) one should considertaking impact phenomena at the snubbers into account, given the very high accelera-tion levels that occur for snubbers that are 100x weaker .

. To gain more insight in the dynamic behaviour of nonlinear dynamic systems, it wouldbe very helpfull if DIANA-module STRDYN would be provided with the possibility tomake plots of the deformed geometry of the system . Once this backtransformation ofthe reduced model to original model is possible, also stresses and strains of a nonlineardynamic system can be analysed .

Appendix A

Eigenmodes

39

MODELa . Z

1" free interface eigenmode of the supported linear structure . X--~r

The snubbers (zero stiffness) are also visible, to improve Eye pa,nt-the visibility of the 3D motion of certain parts . Yà Z :~~~É+~~

Z= 1 .000E+00Rotation: 120 .0 deg

Analysis type : EIGENMode nr: 1Frequency: 3.216E+01

NODAL DATAResult :DISPU TOTAL TRANSL

Axes : GLOBALExtreme values:mox= 1 .005E+00min- 0 O00E+00Deform. x 2 .454E-01

Main motion : Yoke swings in x-z plane

DIANA6t ~" ga00.pi c2-Jul-96 13 .42

MODEL

b . z

2nd eigenmode .-4-"

Eye point :, X= 2.OODE+00

Ym 2 .000E+00Z= 1 .000E+00Rotation : 120.0 deg

Analysis type: EIGENMode nr• 2Frequency : 4.263E+01

NODAL DATAResult :DISPU TOTAL TRANSL

Axes: GLOBALExtreme values:mox= 1 .055E+00min- O .OOOE+00

Deform . x 2.140E-01

Main motion : Yoke swings in x-y plane

DIANA6•' ~e

"'001 .pt~Je2-J.1-96 13 :42

40

a .

b .

3rd eigenmode

4th eigenmode

QIIII61 ti~t1 -I]

MODELz

Eye point:X- 2.000E+00Y= 2 .OOOE+00Z- 1 .000E+00

Rotation : 120 .0 deg

Analysis type: EIGENMode nr: 3Frequency: 7.555E+01

NODAL DATAResult :DISPtA TOTAL TRANSL

Axes: GLOBALExtreme values:,ax= 1 .008E+00min- O.OOOE+00Deform. x 2 .955E-01

DIANA 6teigaa3 .p í~L Jc2-Jul-96 13 :42

MODEL

z

Eye point :X- 2.000E+00Y= 2 .000E+00Z= 1 .000E+00Rotation : 120.0 deg

Analysis type: EIGENMode nr : 4Frequency : 9.933E+01

NODAL DATAResult :DISPIA TOTAL TRANSL

Axes : GLOBALExtreme values:max= 1 .000E+00min= 0 OOOE+DO

Deform . x 2.427E-01

DIANA:-' ~epa03.Vpe3tJN-96 13 :42

aIE!a

41

5th eigenmode

z

Eye po nt :X- 2.000E+00Y- 2 000E+00Z= 1 .000E+00Rototion : 120.0 deg

Analysis type: EIGENMode nr : 5Frequency: 1 236E+02

b.

6th eigenmode

MODEL

NODAL DATAResult:DISPLA TOTAL TRANSL

Axes : GLOBALExtreme values :max= 1 .000E+00min= O,OO0E+00

Deform, x 3 .034E-Oi

DIANA"elgaa4 pie2-Ju1-98 13-42

MODELZ

Eye point:X- 2 .000E+00Y= 2 000E+00Z- 1 .000E+00Rotatwn: 120 .0 deg

Anolysis type : EIGENMode nr: 6Frequency: 1 .324E+02

NODAL DATAResult:DISPLA TOTAL TRANSL

Axes: GLOBALExtreme values:max= 1 .000E+00min : 0 O00E+00Deform. x 3 .034E-01

42

DIANA:,'e~ga05.pWe4tJul-98 13:42

a .

7th eigenmode

MODELZ

Eye point:X~ 2.000E+00Y= 2 .000E+00Z_ 1 .000E+00

Rotation : 120 .0 deg

NODAL DATAResult :DISPLA TOTAL TRANSL

Axes: GLOBALExtreme values:max= 1 .000E+00mine 0.000E+00Deform . x 2 .427E-01

DIANA:-' ~'i9006.01 C2-Jv1-96 13 :42

Analysls type: EIGENMode nr: 7Frequency. 1 .354E+02

Zb .

8th eigenmode

MODEL

Eye point :X= 2.000E+00Y= 2 .000E+00Z= 1 .000E+00Rotation : 120.0 deg

Anolysls type: EIGENMode nr 8Frequency: 1 .574E+02

NODAL DATAResult :DISPLA TOTAL TRANSL

Axes: GLOBALExtreme values:max=e 1 .039E+00min= 0 .000E+00

Deform . x 2 .142E-01

DIANA 6 .1e . yoo 7 o;e

Jul-96 13 .42

43

a.

9th eigenmode

MODELz

Eye point :X- 2 .000E+00Y= 2.000E+00Z= 1 OOOE+00

Rotation : 120.0 deg

Analysts type: EIGENMode nr• 9Frequency: 1 645E+02

NODAL DATAResult:DISPLA TOTAL TRANSL

Axes : GLOBALExtreme values:mox= 1000E+00min - O .OOOE+D0Deform. x 2 .427E-Oi

DIANA67 ~~9008.Pi ~Je2-Ju1-99 13:42

b .

loth eigenmode

MODEL

Z

Eyi point :X- 2 .000E+00Y= 2.000E+00Z= 1 .000E+00Rotation : 120 0 deg

Analysis type: EIGENMode nr : 10Frequency : 1 .676E+02

NODAL DATAResult:DISPLA TOTAL TRANSL

Axes: GLOBALExtreme values :mox= 1 .000E+00min- 0 .000E+00Deform. x 2 .427E-01

44

0

DIANA"eigaa9.Pic2-Ju1-96 13:42

Appendix B

Linear dynamics

45

.

eigenfrequencies [Hz]no. 1692 dof 5894 dof

1 32.751 32.1612 42.648 42.6333 75.487 75.5524 105 .47 99.3275 129 .82 123.636 140 .07 132.357 144.53 135.398 154.93 157.439 170 .02 164.47

10 173 .68 167.5611 188 .56 186.2212 209 .66 203.5513 213.80 207.5414 245.79 244.7315 246.06 245.2516 269.96 275.35

Table B.1: Eigenfrequencies of the base-supported linear model before and after meshrefine-ment

no . k=0 Nm-i k=5 .10' Nm-1 k=1 .101 Nm-11 32 .161 42.543 42 .6512 42 .633 48.024 57.9043 75 .552 79.759 80 .9274 99 .327 100.29 100.485 123.63 142.86 147.556 132.35 159.31 162.237 135.39 171.07 179.618 157.43 199.59 201 .229 164.47 205.44 215.82

10 167.56 227.24 231 .4411 186.22 340.43 437.0012 203.55 425.52 486.0013 207.54 480.03 496.0314 1 1 244.73 501.50 574.65

Table B .2 : Eigenfrequencies [Hz] of the supported system with linear springs for three valuesof k

46

1 z 1 m.x [ml ifcnod 1 a . 1z 1max [m] ifcnod 2 b .wo

10l k.S.OE4Ia1 .0E5 NW'

104

10°

10l

104

25 50 75 100 125 fax [HZ] 25 50 75 100125 fax [HZ]

I z1 max [ml ifCnOd 3 c . ~ z 1max [m] ifcItOd 4 d.

10a 104

„

- -i~1 !

" -;r"- --

104

10$ I

25 50 75 100125 fex [NZl

25 50 75 100 125 fax [Hz]

1 z 1 Max [m] ifcnod 5 e . I Z I max [m] ifcnod 6 f.

104104

104

10{ 2550 75 100 125 fax [HZ]

1 125 50' 75 100 125 fax [HZ]

~ z I m.x [m] ifCnOd 7 g.

-- -- k 0 Nm'110'

k 5•1E »k 1 .l g: »

10,1

10125 50 75 100

125 fex [HZl

Figure B .1: Linear dynamic analyses for three spring stiffness values

47

48

Appendix C

Frequency responses, strongly nonlinear

49

I Z I max [m]

10 -3

50

ifcnod 2

75 100

a .

125 fa. [Hz]

Figure C .1 : Frequency responses of ifcnod 2 and 3, ksnub = 1 .10' Nm-1 .

50

I Z I max Im]

ifcnod 4

10-3

10-a

75

I Z I max Im]

50

ifcnod 5

10-3

25 50 75

I100

100

a.

125fex [Hz]

b.

125 fa, [Hz]

Figure C .2: Frequency responses of ifcnod 4 and 5, ksn„b = 1 .10' Nm-1 .

51

I Z I max [m]

ifcnod 7

10-3

-~ . . ..1w1 1 1 1 1

25 50 75 100 125 fax [Hz]

Figure C .3: Frequency response of ifcnod 7, k$nub = 1• 105 Nm-1 .

52

Appendix D

Demonstration internal excitation

Note : In this report the term "internal excitation" is not used as the cause for superharmonicresonance .

The harmonic solution at 20 .49 Hz will be investigated. The attention has been focused on3

ifcnod 4 and 5, the ends of snubber 3 (fig .D .1) . For x = 0, át3 does not exist (section 3 .5.1) .

Snubber 3 can be described with x3(t) . Contact takes place at t:

(D.1) x3(0 := z5 - z4 = 0

Figure D .2` gives x3(t) . For t E t, z5(t) (fig .D.2b) should be not-differentiable . However inset2 of fig .D .2b shows a bump at t = t* ~ i. This is due to "internal excitation" : At t* snubberS =1, 2 or 4 has state

(D.2) XS = 0

This demonstrates the unpredictiveness of this system .

Resuming: In case a system contains one one-sided spring, all non-differentiable points of z(t)follow from z(t) = Za - Zb = 0 at i(i). If more one-sided springs are involved in one structure,this is not true .

I I I I - _ I

a .

-1 .0E-3 -5.0E-4 -3 .2E-10 5.0E-4 1.0E-3 1 .5E-3displacement [m]

ifcnod 5ifcnod 4

O.OEO 1.0E-2 2.0E-2 3.0E-2 4.0E-2

b .

5.0E-2 time [s]

Figure D .1: Ifcnod 5, feX = 20 .49 Hz, harmonic solution, n,. = 2000

54

velocity [ms']

0.20

0 . 10

0.00 ---------------------

-0 .10

-0 .20

0.OE0

zs'z4 [MI

2.0E-4

1 .5E-4

1 .0E-4

5.0E-5

0.OE0

I1 .0E-2 2.0E-2

,3.0E-2

,4 .0E-2

a.

S.OE-2 time [s]

0.0E0 1.0E-2 2.0E-2 3.0E-2 4.0E-2 5.0E-2

Figure D .2: Stable. h,~~monic solution at feX = 20 .49 Hz, n7 = 2000 .a. • z l~5b . . z5(t)C . : X3(t) = Z5 - Z4

C .

time [s]

55

56

Appendix E

1/17 S ubharmonic solution

For both figures, Adams integration has been applied, with an accuracy of 9 significant digits .

Figure E.2 reveals that the somewhat exotic shape in figure E .1 does not originate from somerepeating numerical integration error, but from a 1/17 subharmonic solution .

Because of the very stretched out axis in fig.E.2, round-off errors are visible : that causes thejumping behavior . Considered well, the points in the Poincaré section are all the same :

IZ1,max - Zl,min) = 1.200•10-9 m

1 Z1,max - Z1,minl = 1 .342• 10-s MS-1

The stable harmonic in fig.E.2 converges very slowly, because the maximum value of one ofthe Floquet multipliers is almost one :

fex IAImax34.700000 ?34.700157 0.93234.703278 1 .591

ResponseStable harmonicStable 1/3 subharmonicquasi-periodic behavior (?)

range34.69 - 34.71 Hz34.72 - 34.76 Hz34.77 - 34.83 Hz

notation for µ000

57

velocity [ms']

-0.05750

-0.05800

-0.05850

-0.05900

-0.05950

-0.06000-1 .1485E-3 -1.1480E-3 -1 .1475E-3 disp. [m]

Figure E.1 : Poincare section PQOO(Ó) . It looks like quasi-periodic behavior locked on a 1/17subharmonic at 34.827 Hz .

velocity [ms']200 T„

600 T ex

6.118f1452E-3

i-1 .1452E-3 -1 .1452E-3 -1.1452E-3

800 T„

dis p. m

Figure E.2: Poinca,ré section of stable harmonic solution at feX = 34 .700 Hz., ifcnod 1 :

P2'0',)' ( s (3000TeX) If, = 34.827) , Psóó ('7)•

58

3(µ) square : f„ = 34 .7126 Hzi diamond : 34.7251 < f„ < 34 .7501 Hz

't circle :34 .77 51 < f„ < 34 .8270 Hz

1 .0

M

0.5

0

0~10

~ 914

DDM

w

(6~/~J

0.0 L•/~ . ./~. ! ~ ah. ~~AR\/Afl~ ~l1 .

-1 .5 -1 .0 -0.5 0.0 0.5®®

Figure E .3 : Floquet multipliers for 34 .69 < fe7C < 34.83 Hz .

dz/dt [ms']

0.150

0.100

0.050

0.000

-0.050

-0.100

-0.150-1 .0000E-4 7.2760E-1 2

9fcnod 5

1 .0000E-4

b .

2.0000E-4 Z [m]

Figure E .4: Phase portraits of the low-amplitude unstable harmonic and high-amplitude stable1/2 subharmonic solution at feX = 88 .00 Hz .

59

60

Appendix F

Strongly nonlinear

velocity [ms'] a . velocity[ms'] b.

0 .150

0.100 0 .100

0 .050 0 .050

0 .000 0 .000

>K

-0.050 -0 .050

-0.100 -0 .100

-0.150 -0 .150 -,--1 .0000E-4 7.2760E-12 1 .0000E-4 disp. [m] - 1.0000E-4 7.2760E-12 1 .0000E-4 disp. [m]

displacement [m] c . Velocity [ms'] d .

0.1501 .50E-4

0 .1001 .00E-4

0.050S .OOE-5

r

0.0005.09E-11 ;

-0.050-5 .00E-5

-0.100-1 .00E-4 ~

0-0.150 0. . '

0.0000 0.0100 0.0200 0.0300 0.0400 time [S] -1.0000E-4 7.2760E-12 1.0000E-4disp . [m]

acceleration [ms Z] e• acceleration[ms 2] f•

2.OOE2 2 .OOE2

41 .OOE2 1 .OOE2

O.OOEO 0 .00E0

-1 .OOE2 -1 .OOE2 ~~6

-2 .OOE2 -2 .OOE2

0.0000 0.0100 0.0200 0.0300 0.0400 time [s] 0.0000 0.0100 0.0200 0.0300 0.0400 time [S]

a.ZS Z4 [m] g .b.C .

2 .50E-4 d .2 .00E-4 e

f1 .50E-a

g1 .00E-4

5 .00E-5

3.64E-12

-5.00E-50.0000 0.0100 0.0200 0.0300 0

.0400 time [S]

Figure F.1 : Stable 1/4 subharmonic solution at feX = 96 .15 Hz

phase portrait ifcnod 4phase portrait ifcnod 5displacement ifcnod 4 (dotted) and 5 (solid}phase portrait ifcnod 4 (dotted) and 5(solid}acceleration ifcnod 4acceleration ifcnod 4 (dotted) and 5 (solid)(compression of) snubber 3 between

d 4 and 5ifcno

62

v.weUN~11

0.050

-0.000

-0.050

Henod 1

a.

k~ ~ ~~ . ~ . . . . ~ . . . __ ~

-1 .00E-4 -5.00E-5 -3.82E-1i 5 .00E-5 1.00E-4, di.pl.c.m .nt [m]

b.

r. ~-5.00E-5 -4 .37E-11 5.00E-5 1.00E-4 1 .50E-4

dhpl .o.m.nt [m]

Figure F .2: Phase portraits of ifcnod 1 and 7(nr=2000) of stable 1/4 subharmonic solutionat feX = 97.80 Hz (flip bifurcation point at 97 .93 Hz on 1/2 subh. branch). On can imaginehow the accompanying unstable 1/2 subharmonic solution looks like .

Nµ)

1 .00

0.75

0.50

0.25

$

f,x 96.808 Hz

~~ f„97.018 Hz

0VA

W11 q

m

~ Sol--Al ~ e

i

190.00 6 é* . i

-1 .0 -0.5 0.0 0.5 1.0 9tW

Figure F .3 : Floquet multipliers for the 1/4 subharmonic solution branch . Between 96.808-97.018 Hz quasi-periodic behaviour has been found .

4,inAAiW_;I

63

velocity [ms']

-1 .00 E-4 -5 .00 E-5 -3.55E-1 1 5.00 E-5 1 .00 E-4

a.

1 .50E-4displacement [m]

Figure F.4: (a) Phase portrait of ifcnod 5 at feX = 96.900 Hz, (b) Enlargement of boxed areafig. F .4

64

a .

velocity [ms']

-0.150

-0.160

-0.170

-0.180

-0.190

K-4.70E-4 -4.60E-4 -4.50E-4 -4.40E-4 -4.30E-4 disp. [m]

b .

4.00E-5 4.50E-5 5.00E-5 diSp•[m ]

Figure F.5 : Quasi-periodic behaviour at feX = 96 .900 Hz, Poincaré sections P26000000(6) of ifcnod3 and 6 respectively.

65

0.50

0.25

1 µ 1 =1 at f.X= 95.51 Hz

PIT

fex0.00

Oh

0 nef 9

-1 .10 -1 .00 -0.90 -0.80 -0.70

1313 0gf e

i

0 0 0 0d c b a

Nµ)

Figure F .6 : Floquet multipliers of the 1/2 subharmonic solution branch: Multipliers markedwith the same letter correspond to the same frequency. At 95.17 Hz (unmarked), P lJs(µ) 54 0breaks through the unit circle : Neimark bifurcation . Following the Floquet multipliers forincreasing frequency, at feX=95 .51 Hz (between d and e) (inverse) Neimark and flip bifurcationare met almost simultaneously . This flip bifurcation initiates the 1/4 subharmonic branch .

~3m&b

66

velocity [ms']

-0 .100

-0 .125

-0 .150

-0 .175

g .

-0 .200

-8 .00E-5 -6.00E-5 - 4.00E-5 -2.00E-5 1 .64E-11 disp- [m]

velocity [ms']

IZj Imax [ml

1 .421 .10-4

7.635-10-47.424- 10-41 .855-10-42.527-10-41 .474-10-42.153•10-4

Figure F.7: Poincaré sections P620'0'0"(6) at fex=99.136 Hz, Adams integration with 9 significantdigits: Chaos. Estimations of the maximum of the displacement amplitudes of the vario Vifcnods are given in the table above .

b .

I displacement [m]

~1 .00E-4 ~ ~

5.00E-5

3 .55E-11

-5 .00E-5

-1 .00E-4

Pa

ifcnod 1

Í

i4

b

20.200 20.300

displacement [m]

2.50E-4

2.00E-4

1 .50E-4

1 .00E-4

5.00E-5

9.09E-1 2

-5.00E-5

i

á

20.200 20 .300

Illl r' ~~~20.400 20.500 20.600

time [a]

ifcnod 5

i20.400 20.500 20.600

N ~tn~~wl9ppir~ra

a .

C .

time [s]

displacement [m]

1 .00E-4

5 .00E-5

3.55E-11

-5 .00E-5

-1 .00E-4

L

01

w

20.190

h I'8

18

displacement [m]

2 .50E-41-

2 .00E-4

1 .50E-4

1 .00E-4

5 .00E-5

9.09E-12

-5.00E-5 E

.y20.190

ifcnod 1

M

v

V

20 .220 20.250

ifcnod 5

Iq20.220

u20.280

a

20 .250

d .

20.280 time [s]

Figure F.8 : Some time histories and phase portraits of the chaotic response at 99 .136 Hz .The time between two o-symbols is one excitation period. (Begin time was 2000Te7C . AdamsMegration of another 50TeX7 and saving the systems' state 200 times per T, (NnT=200 incase solution would be periodic) resulted in these figures .)

b .

time [s]

Appendix G

Frequency spectra

This appendix handles the frequency spectra of the response for three different excitationfrequencies :1. fe,t = 58.000 Hz : Harmonic system response2. fe,, = 96.900 Hz : Quasi-periodic system response3. fe,, = 99.136 Hz : Chaotic system response

Fast Fourier 1'ransformation

The frequency spectra have been determined using FFT (De Kraker ('92)[7]) . For bettercontrol, the approach is as follows : The response in terms of displacements is sampled with asample frequency

(G.1) fN=c•feX

with c> 2 and c E N to anticipate on aliasing and signal leakage respectively . For N spectrallines, DIANA saves the displacements at

(G.2) tg=to + n(1/fN), n=1, . . .,N

This output (9 significant digits) will be provided to MATLAS's FFT-algoritme, and resultsin a frequency spectrum of the sampled response .

For the three cases, the following settings are used :

1 . fe,, = 58 .000 Hz, fN = 8• 58 .000 = 464.00 Hz. As 1024 spectral lines are desired, the to-tal sample time is tg(n=1024) = 2 .21 s N 128 TeX . The fold-frequency is 464/2=232 Hz .This results in an effective frequency spectrum with 512 spectral lines with the range 0-232 Hz : Of = 512/232 = 0.4531 Hz .(figures G . la-b)

2 . fe,, = 99.136 Hz, fN = 7• 99 .136 = 693 .95 Hz, te(n=2048) N ±292 Te,, ,resolution Of = 693 .95 _ 346.98 = 0.3388 Hz2048 1024(figures G .1°-a)

3 . fe7C = 96 .900 Hz, fN = 7• 96.900 = 678.30 Hz, t,, (n=2048) - ±293 Te% ,resolution Of = 678 30 = 339 .15 = 0 .3321 Hz

2048 1024

(figures G.le-f)

69

Iz l [ml a . Iz,l [ml b .,ifcnod 1

1 .40E-4 ifCnOCf 1 10,

1 .20E-4lol

10'1 .00E-4

10''

8.00E-5 10+

6.00E-5 104

4 .00E-5 7010

10-"2 .00E-5

10'u0 .00E0

0 50 100 150 200 f [HZ] 0 50 100 150 200 f [ j=]

Iz,l [ml c . Iz,l [ml d .2 .00E-4 iÍCnOCJ 2 ifCnOd 2

1o,

1 .50E-4104

1 .00E-410°

S .OOE-510-'

0.O0E050 100 150 200 250 300 f [H Z] 50 100 150 200 250 300 f [HZ]

I z, l[ml e . I z, l[ml f.2.00E-5 ifCnOCf 1

104

10`1 .50E-5

10l

1cO8E-5 10-r

10'

5 .00E-610'

0.00E0 10'"50 100 150 200 250 300 f [HZ] 50 100 150 200 250 300 f[HZ]

fig .a .b .C .d .e .f.

feX [HZ l58 .00058 .00096 .90096 .90099 .13699 .136

Figure G .1 : Frequency spectra of harmonic, quasi-periodic and chaotic response .

responseharmonicharmonicquasi-periodicquasi-periodicchaoticchaotic

scalelin .log .lin .log .lin .log .

70

Appendix H

Comparison nonlinear / linear

71

IZlmsx[m]

10° S"

104

25

I 2 I mex [m]

10°

104

10{

r

25

125 }sx [HZ]

1,

,50

'rfcnod 3 c

I75

~

I z 1,..„[m] ifcnod 2 b.

104

Ili

104

25,

7550 100

100

ifcnod 5e.

W 75 100 125 fax[HZ]

ksnub = 1•105 Nm-1k = 0 "k = 5•104 "k = 1•105 "

125 }sx [HZ]

Figure H .1: Comparison of linear dynamic analyses (k=0, 104, 105 Nm-1) and nonlineardynamic analyses (ksnub=105 Nm-i)

Appendix I

Path following frequency ranges

73

Harmonic branches127 .537120 .00099.110899.110897.000082.389180.500074.939474.939470.798268.000062.138260.000053.742351.000049.778243 .000037.919934.827233.000025.341825.000024.000023.000020.4877

120.00099 .110899.107198.399982.511580.510076.127774.000070.914268.000061 .978960.000053.739151 .000049.692443.000037.919935.482733.000026.145124 .000024 .918323 .038118 .619020 .0000

1/2 Subh.99.999498.260296.510092.280097.900090.305890.000087.694786.648486.900081 .900081.162980.000076.061975.9000

branches96.530098.099992.330091.786891.785190.100087.686287.099982.100086.427981.162980.099976.696675.900075.7536

1/4 Subh. branches97.1500 -> 97.090797.0928 f- 96.818496.8084 - 96.031497.9290 - 97.150096.0752 - 96.000095.9800 -> 91 .7868

Table 1.1 : Start and terminal frequencies of the path following efforts (ksnub=105 Nm-1) . Thearrows indicate the direction of the pf process . At almost all terminal points, the stepsizevk < 10's

In the amplitude-frequency plots for ksnub = 105 Nm-1, all solutions branches within thefrequency ranges of table I .1, are collected .

Some figures :

Each amplitude-frequency plot of ifcnod j = 1, . . .7 holds 3103 values I zi I ma', :

Harmonic solution branch : 1608 points1/2 Subharmonic " " : 1152 , 1

1/4 Subharmonic " " : 343 " +Total : 3103 points (/plot /ifcnod)

This single value J zj Im. has been drawn from the n,-column zj , representing a time-discretizedperiodic solution for ifcnod j, on an average of ii, ;zt~ 1500. This solution zj just a subset ofthe total solution, for a given frequency . For each point (fex, kilmax) the system h(z) = 0 hasto be solved. Column z has length nq x fir = 14 x 1500 = 21000. (21000 equations, 21000unknowns)

74

Appendix J

Finding a periodic solution: example

An example will illuminate the applied approach for finding a"diíficult" periodic solution .Difficult in the way that a standard start solution converges very slowly (too slow for practicalreasons) to a periodic solution . Then, numerical integration is applied to find a better startsolution .

Consider the case: k$nub = 105 Nm-1, nq = 14 dof (fc = 125 Hz) .

At fe1C = 88 Hz two solutions are possible : an unstable low-amplitude harmonic solution and astable high-amplitude 1/2-subharmonic . For t=0 the initial state is specified : so =[Ot 0t]tIntegration of 300Te7C gives a Poincaré section PZO°(so), assuming the transient has died outin 200 excitation periods . Consider just the states of one ifcnod . These states in this Poincarésection are grouped in two reasonably converged point collections indicating the trajectory ofthis ifcnod is periodic with two times the excitation period (_ $$ s). This means the responsof the whole system is 1/2 subharmonic .

This 1/2-subharmonic has to be provided to the "two-point boundary value problem Solver",

in terms of a npnT-column zo . The systems' trajectories are integrated for two more excitation

periods: 300 < T< 302. Choosing nT = 2000, the state s, will be saved at -r = 300 -I-

2i/2000, i = 0, . . . , 2000 - 1. Collecting these 2000 states in a column gives us the desired

start solution zo. This zo led after just 3 iterations to the stable 1/2-subharmonic solution

zf--sa Hz •

For phase portraits of the periodic solutions at 88 Hz, see figure E .4 .

75

76

Appendix K

Numerical data of the model

Due to symmetry of the geometry of the system, and the symmetry of excitation only onehalf is modelleá. The total system mass is 22 .03 kg, the mass of the FEM model is 22.03/2 =11 .02 kg .

K.1 Solar Panels

Modelled using plate bending elements (Q20SF) [2] .

The system consists of 2 solar panels (l x b x H = 2.1 x 1 .9 x 22 .36 . 10-3 m). They have asandwich structure: The center is a so-called honey-comp (thickness h = 22 mm, E := 0) . Itis covered on both sides by so-called skin plates (thickness s = 0.18 mm, E= 150•109 Nm-2) .

The I of the whole (composed) panel is calculated as follows (formula provided by Fokker) :

(K.1) 1 = 2 h2 t= 4.356-10-8 m3

Skin plates can not be compressed in the plane of their surface . A representative thickness tRof the plate bending elements can be calculated by :

(K.2) 2tR = 2 h2 t ~ tR = 8.0554•10-3 m

and

(K.3) IR = I b = 8.2764• 10-$ m4

The density of the panels can be calculated using the given m = 2.2 kgm-2 :

(K.4)M m l b 2 .2

= 98.3899 kgm-3PV H l b 22 .36-10-3

The input for the datafile of DIANA concerning the solar panels is :

YOUNG 150•109POISON 0.3

DENSIT 98.3888THICK 8 .0554•10-3INERTI 8 .2764• 10-8

77

K.2 Edgemembers

Modelled using beam elements (L12BE) [2] .

Given : E = 15 GPa, A = 2.3•10-5 m2, I = ZJ = 2•10'9 m4. These edgemembers are massless .

YOUNG 15•109POISON 0.49

DENSIT 1CROSSE 2.3•10'5

INERTI 2.0•10'9 2.0•10'9 4.0•10'9 (local 1,., Iyy, and I,,)

K.3 Holddown systems

Modelled using beam elements (L12BE) [2] .

K.3 .1 Holddown between satellite and panel 1 (lower panel)

Given: E = 70 GPa, A = 100 . 10-6 m2, I= 2 J= 7 .0 • 10'8 m4. No mass specified .

YOUNG 70 •109

POISON 0.49DENSIT 1

CROSSE 100•10'6IN ERTI 7.0- 10-s 7.0-10-8 14.0-10-8

ZAXIS 0. 1 . 0 .

K .3.2 Holddown between panel 1 and panel 2

Given : E = 70 GPa, A = 62-10-6 m2, I = 2 J = 2 .2-10' m4 . No mass specified .

YOUNG 70•109POISON 0.49DENSIT 1CROSSE 62•10'sINERTI 2.2-10-' 2.2-10-8 4.4-10'ZAXIS 0 . 1. 0 .

K.4 Yoke

Given: E= 200 GPa, A = 1 .2 . 10'4 m2, I= 'J = 3• 10-s m4, and m = 1 kgm-' . Density iscalculated using :

(K.5)M _ m ly _ 1

P=V Aly 1 .2•10-4 = 8.3333 . 103 kgm-3

YOUNG 200•109POISON 0 .49DENSIT 8 .3333•103CROSSE 1 .2•10'4INERTI 3 .0•10'8 3 .0•10-8 6.0•10-8

78

K.5 Joints

The properties of the beams b3-bc, bc-c6, c3-cd, and cd-d3 .

Given: E = 70 GPa, A= 200-10' m2, I = 2 J = 5 .10-' m4, and m = 2.5 kgm-1 :

= 12.5•103 kgm-3(K.6) p = m-x

YOUNG 70•109POISON 0.49DENSIT 12 .5•103CROSSE 200•10-6IN ERTI 5.0-10-9 5 .0-10-9 10.0-10'

79

80

Appendix L

User-force subroutines

L.1 Definition external load

SUBROUTINE USRLOD( TASK, FNR, DESV, TIME, DESNR, LOD , DDES )CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .COPYRIGHT (C) TNO-BOUWC . . .C . . . EXTENDED WITH A PRESCRIBED BASE ACCELERATION (DESV(5)) !!!

C . . .DOUBLE PRECISION TWOPIPARAMETER ( TWOPI=6 .2831853D0 )

C

C

C

DOUBLE PRECISION LOD(4), TIME, DDES(8), DESV(*)INTEGER DESNR(*), FNRLOGICAL TASK(*)

DOUBLE PRECISION PHI

CALL RSET( O .DO, LOD, 4)CALL RSET( O .DO, DOES, 8)

CIF ( DESV(9) NE . O .DO ) THEN

PHI = DESV(9) / 360 .DO*TWOPIELSE

PHI = O .DOEND IF

C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IF ( TASK(1) ) THEN

IF ( FNR EQ . 2 ) THENLOD(2) = -DESV(S) * COS(TWOPI*DESV(2)*TIME + PHI)/

$ (TWOPI*TWOPI * DESV(2)*DESV(2))LOD(3) = DESV(5) * SIN(TWOPI*DESV(2)*TIME + PHI)/

$ (TWOPI*DESV(2))LOD(4) = DESV(5) * COS(TWOPI*DESV(2)*TIME + PHI)

END IFEND IF

81

C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IF ( TASK(2) ) THEN

CALL RSET( O .DO, DOES, 4)IF ( DESNR(1) EQ . 2 ) THEN

IF ( FNR EQ . 2 ) THENDDES(2) = 2 .DO*DESV(5)*COS(TWOPI*DESV(2)*TIME+PHI)/

$ (TWOPI*TWOPI * DESV(2)*DESV(2)*DESV(2))DDES(3) = -DESV(5)*SIN(TWOPI*DESV(2)*TIME+PHI)/

$ (TWOPI * DESV(2)*DESV(2))DDES(4) = O .DO

END IFELSE IF ( DESNR(1) EQ . 5 ) THEN

IF ( FNR EQ . 2 ) THENDDES(2) = -COS(TWOPI*DESV(2)*TIME + PHI)/

$ (TWOPI*TWOPI * DESV(2)*DESV(2))DDES(3) = SIN(TWOPI*DESV(2)*TIME + PHI)/

$ (TWOPI*DESV(2))DDES(4) = COS(TWOPI*DESV(2)*TIME + PHI)

END IFEND IF

END IFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IF ( TASK(3) ) THENCALL RSET( O .DO, DDES(5), 4)IF ( DESNR(2) EQ . 2 ) THEN

IF ( FNR EQ . 2 ) THENDDES(6) = 2 .D0*DESV(5)*COS(TWOPI*DESV(2)*TIME+PHI)/

$ ( TWOPI*TWOPI * DESV(2)*DESV(2)*DESV(2) )DDES(7) = -DESV(5)*SIN(TWOPI*DESV(2)*TIME+PHI)/

$ ( TWOPI * DESV(2)*DESV(2) )DDES(8) = O .DO

END IFELSE IF ( DESNR(2) EQ . 5 ) THEN

IF ( FNR EQ . 2 ) THENDDES(6) = -COS( TWOPI*DESV(2)*TIME + PHI )/

$ ( TWOPI*TWOPI * DESV(2)*DESV(2) )DDES(7) = SIN( TWOPI*DESV(2)*TIME + PHI )/

$ ( TWOPI*DESV(2) )DDES(8) = COS( TWOPI*DESV(2)*TIME + PHI )

END IFEND IF

END IFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

END

82

L .2 Definition one-sided spring

SUBROUTINE NDSPTR( TASK, ELMNR, N, DESNR, DESVL, TIME,$ DISL, FORL, DFDI, DFDII, STIFF, DAMP,$ DUM1, DUM2, DUM3, DUM4, DUMS )

CCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COPYRIGHT ( C) TNO-BOUWC . . . CALCULATE INTERNAL FORCES . . . ETC .C . . . FOR THE 3-DIM . 2-NODE SPRING ELEMENT 'SP2TR'C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CC . . . DESVL(1)C . . . DESVL(2)C . . . DESVL(3)C . . . DESVL(4)C . . . DESVL(5)C . . . DESVL(6)C . . . DESVL(7)C . . . DESVL(8)C . . . DESVL(9)C

C

CC

exitation frequencybacklashstatic load C = 0 )amplitude of base acceleration (= 2g )one-sided dampingone-sided stiffness

phase

DOUBLE PRECISION TIME, DESVL(*), DISL(*), FORL(*), DFDI(*),$ DFDII(*), STIFF(*), DAMP(*), DUM1(*), DUM2(*),$ DUM3(*), DUM4(*), DUMS(*)INTEGER ELMNR, N, DESNR(*)LOGICAL TASK(*)

DOUBLE PRECISION TDELTA, TVELTA, TFORCE, SPSTIF, SPDAMP

TFORCE = O .DOSPSTIF = O .DOSPDAMP = O .DOTDELTA = DISL(2) - DISL(1)TVELTA = DISL(4) - DISL(3)IF ( DESVL(7) NE . O .DO AND . TDELTA LT . -DESVL(3) ) THEN

CC . . . Lineair stiffnessC

TFORCE = TFORCE + DESVL(7) *( TDELTA + DESVL(3) )$ + DESVL(6) * TVELTA

SPSTIF = SPSTIF + DESVL(7)SPDAMP = SPDAMP + DESVL(6)

C

CEND IF

83

IF ( TASK(1) ) THENFORL(1) = -TFORCEFORL(2) = +TFORCE

END IFC

IF ( TASK(2) ) THENCC . . . Setup JacobianC

C

STIFF(1) = +SPSTIFSTIFF(2) = -SPSTIFSTIFF(3) = -SPSTIFSTIFF(4) = +SPSTIF

DAMP (1)DAMP (2)DAMP(3)DAMP(4)

END IF

+SPDAMP-SPDAMP-SPDAMP+SPDAMP

C

C

C

C

C

C

IF ( DESNR(1) NE . 0 ) THEN

TFORCE = O .DO

IF ( DESNR(1) EQ . 3 ) THENIF ( TDELTA LT . -DESVL(3) ) TFORCE = DESVL(7)END IF

IF ( DESNR(1) EQ . 6 ) THEN

IF ( TDELTA LT . -DESVL(3) ) TFORCE = TVELTAEND IF

IF ( DESNR(1) EQ . 7 ) THENIF ( TDELTA LT . -DESVL(3) ) TFORCE = TDELTA + DESVL(3)END IF

DFDI(1) = -TFORCEDFDI(2) = +TFORCE

END IF

END

84

Bibliography

[1] D.H . van Campen . Niet-lineaire Dynamica. Syllabus 4661, Eindhoven University of Tech-nology, Eindhoven, The Netherlands, 1991 . (In Dutch)

[2] DIANA User's Manual, release 6.0. TNO Building and Construction Research, Delft, TheNetherlands, 1996 .

[3] R.H .B . Fey . Steady-state Behaviour of Reduced Dynamic Systems with Local Nonlinear-ities. Ph.D . thesis, Eindhoven University of Technology, Eindhoven, The Netherlands,1992 .

[4] R.H .B . Fey, D .H. van Campen, A. de Kraker . Long Term Structural Dynamics of Me-chanical Systems With Local Nonlinearities . Journal of Vibration and Acoustics, Vol .118, pp. 147-153, 1996 .

[5] Robert C . Hilborn . Chaos and Nonlinear Dynamics, an introduction for scientists andengineers . Oxford University Press, New York, 1994

[6] M .K .J. de Jager . Chaotische Dynamica En Fundamentele Werktuigbouwkunde . Thesisfor the engineering degree TUE :WFW91 .073, Eindhoven University of Technology, Eind-hoven, The Netherlands, 1991 . (In Dutch)

[7] A. de Kraker . Numeriek-Experimentele Analyse van Dynamische Systemen . Syllabus4668, Eindhoven University of Technology, Eindhoven, The Netherlands, 1992 . (In Dutch)

[8] T .S . Parker and L .O . Chua . Practical Numerical Algorithms for Chaotic Systems .Springer-Verlag, New York, 1989 .

[9] W.H . Press, B .P. Flannery, S .A. Teukolsky, W .T. Vetterling . Numerical Recipes, The Artof Scientific Computing . Cambridge University Press, 1986 .

[10] Rudiger Seydel. From equilibrium to Chaos, Practical Bifurcation and Stability Analysis .Elsevier Science Publishing Co ., Inc ., 1988 .

[11] J.M.T . Thompson and H .B . Stewart . Nonlinear dynamics and Chaos. John Wiley & SonsLtd., 1986 .

[12] E.L.B . van de Vorst . Long Term Dynamics and Stabilization of Nonlinear MechanicalSystems. Ph .D . thesis, Eindhoven University of Technology, Eindhoven, The Netherlands,1996 .

[13] E.L.B . van de Vorst, D .H . van Campen, A . de Kraker, and R .H .B . Fey. Periodic Solutionsof a Multi-dof Beam System with Impact . Journal of Sound and Vibration, Vol. 192, pp .913-925, 1996

[14] E.L.B . van de Vorst, D .H. van Campen, A. de Kraker, and R.H.B . Fey. ExperimentalAnalysis of the Steady-State Behaviour of Beam Systems with Discontinuous Support .Meccanica, Kluwer Academic Publishers, Vol . 31, pp . 293-308, 1996 .

[15] M .W.J.E. Wijckmans . Dynamic Analysis of Idealised Solar Panels with a Bilinear Snub-ber. Thesis for the engineering degree TUE:WFW95 .071, Eindhoven University of Tech-nology, Eindhoven, The Netherlands, 1995 .

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