A C
ONTROLL ABILITY
A
PPROACH FOR
R
ESONANT
C
OMPLIANT
S
YSTEMS
A
PPLIED TO A
F
LAPPING
W
ING
M
ICRO
A
IR
V
EHICLE
D
ESIGN
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 22 maart 2016 om 10.00 uur
door
Hugo Jacobus P
ETERS
werktuigbouwkundig ingenieur geboren te Woerden
promotor: prof. dr. ir. F. van Keulen copromotor: dr. ir. J.F.L. Goosen Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. ir. F. van Keulen, 3mE, TU Delft, promotor Dr. ir. J.F.L. Goosen, 3mE, TU Delft, copromotor
Onafhankelijke leden:
Prof. dr. C. Bisagni, LR, TU Delft Dr. G.C.H.E. de Croon, LR, TU Delft Prof. dr. ir. W. Desmet, KU Leuven Prof. dr. ir. M. Wisse, 3mE, TU Delft Prof. dr. ir. D.J. Rixen, TU München
Prof. dr. ir. J.L. Herder, 3mE, TU Delft, reserve lid
This work is part of the Atalanta project from Cooperation DevLab and is supported by Point One - UII as project PNU10B24, Control of Resonant Compliant Structures.
Keywords: resonant compliant systems; resonance response; controllability; smart materials; con-trol actuator design; Flapping Wing Micro Air Vehicle (FWMAV).
Front & Back: Schematic of resonance flap response control of a simplified FWMAV design. An electronic version of this dissertation is available at
http://repository.tudelft.nl/ Copyright © 2015 by H.J. Peters ISBN 978-94-028-0071-5
A CONTROLL ABILITY
APPROACH FOR
RESONANT
COMPLIANT
SYSTEMS
A
PPLIED TO A
F
LAPPING
W
ING
M
ICRO
A
IR
V
EHICLE
D
ESIGN
Praise him for his mighty acts: praise him according to his excellent greatness.
C
ONTENTS
Summary xiii
Samenvatting xv
1 Introduction 1
1.1 Background to FWMAV design . . . 2
1.2 Necessity for control . . . 3
1.3 Controllability of resonant compliant FWMAVs. . . 4
1.4 Thesis objective. . . 7
1.5 Thesis outline. . . 7
References. . . 10
I Actively Modifying Resonance Responses 13 2 Eigensolution Modifications 15 2.1 Introduction . . . 16
2.2 Local structural stiffness changes. . . 16
2.3 Local structural damping changes . . . 19
2.4 Modal phenomena . . . 23
2.5 Practical implications. . . 24
2.6 Conclusions. . . 24
References. . . 25
3 Undamped Resonance Modes Modifications 27 3.1 Introduction . . . 28
3.2 Modifications using local structural changes . . . 30
3.2.1 Eigenproblem formulation. . . 30
3.2.2 Control configurations. . . 31
3.2.3 Control actuators and control signal. . . 31
3.2.4 Control power . . . 32
3.2.5 Eigenproblem sensitivity. . . 32
3.3 Repeated eigenvalues. . . 34
3.3.1 Eigenproblem sensitivity in a modal basis. . . 34
3.3.2 Eigenproblem sensitivity for systems with repeated eigenvalues. . . 35
3.4 Effective resonance mode control. . . 37
3.4.1 Resonance mode modification measure. . . 37
3.4.2 Optimization formulation . . . 38 vii
3.5 Numerical example. . . 39
3.5.1 Numerical model and free vibration eigensolutions . . . 39
3.5.2 Control configuration and corresponding projection set. . . 40
3.5.3 Modal contribution for nearly repeated eigenvalues. . . 41
3.5.4 Determining the most effective control patch locations . . . 42
3.6 Conclusions. . . 43
References. . . 44
4 Damped Resonance Modes Modifications 47 4.1 Introduction . . . 48
4.2 Method for damped resonance mode modifications . . . 49
4.2.1 Damped eigenproblem formulation . . . 49
4.2.2 Modulus and argument of a resonance mode . . . 50
4.2.3 Damped eigensolution sensitivities . . . 51
4.2.4 Modulus and argument sensitivities . . . 52
4.2.5 Resonance mode modification measure. . . 52
4.3 Numerical example. . . 54
4.3.1 N DOF mass-damper-spring system. . . 54
4.3.2 Free vibration eigensolutions . . . 54
4.3.3 Projection set for magnitude and phase modifications. . . 55
4.3.4 Most effective locations to modify structural property. . . 56
4.4 Conclusions. . . 57
References. . . 57
5 Response Modifications of Damped Resonant Systems 59 5.1 Introduction . . . 60
5.2 Modal based dynamic response. . . 61
5.2.1 Viscously damped vibration problem . . . 61
5.2.2 Damped eigenvalue problem formulation. . . 62
5.2.3 Frequency Response Function. . . 63
5.2.4 Modal based response in time . . . 64
5.2.5 Polar representation response amplitude . . . 65
5.3 Resonance response modifications . . . 66
5.3.1 Systems with distinct or repeated eigenvalues. . . 66
5.3.2 Design or control variables. . . 66
5.3.3 Response amplitude sensitivities. . . 67
5.4 Effective resonance response control . . . 71
5.4.1 Response amplitude modification measure . . . 71
5.4.2 Design of resonance response control . . . 72
5.5 Numerical example. . . 74
5.5.1 Discrete system and free vibration eigensolutions. . . 74
5.5.2 Resonance response due to excitation . . . 77
5.5.3 Normalized modulus projection set . . . 78
5.5.4 Determining the most effective control variable: System A and Sys-tem C . . . 78
CONTENTS ix
5.5.5 Effectiveness of control variables as a function of the system
prop-erties. . . 80
5.6 Conclusions. . . 82
References. . . 83
II Inducing Structural Property Changes 87 6 An Inventory to Structural Changes 89 6.1 Introduction . . . 90
6.2 Methods to induce structural changes . . . 91
6.2.1 Mass distribution changes. . . 91
6.2.2 Temperature stimuli. . . 91
6.2.3 Introduce stress distributions . . . 92
6.2.4 Smart fluids . . . 93
6.2.5 Effective length changes . . . 93
6.2.6 Pneumatics . . . 94 6.2.7 Cross-sectional adjustments. . . 95 6.2.8 Electrostatic softening. . . 97 6.2.9 Piezoelectric materials. . . 97 6.2.10 Electroactive polymers. . . 98 6.2.11 Concluding remarks . . . 98
6.3 Qualitative comparison between methods . . . 98
6.3.1 Selection criteria. . . 98
6.3.2 Qualitative comparison and resulting promising candidates. . . 99
6.4 Conclusions. . . 101
References. . . 102
7 Electrostatic Sticking to Modify the Passive Pitching Motion 109 7.1 Introduction . . . 110
7.2 Passive pitching flapping motion . . . 111
7.2.1 Flapping wing design . . . 111
7.2.2 Wing kinematics and passive pitching. . . 112
7.3 Analysis of electrostatically controlled hinge . . . 113
7.3.1 Proposed elastic hinge design . . . 113
7.3.2 Voltage-induced stresses between stacked layers . . . 114
7.3.3 Behavior of the active hinge during large deflections. . . 115
7.3.4 Voltage-dependent hinge properties. . . 118
7.4 Equation of motion of passive pitching motion. . . 119
7.5 Experimental and analytical analysis . . . 121
7.5.1 Realization of wing with active hinge. . . 121
7.5.2 Experimental setup . . . 122
7.5.3 Experimental results. . . 123
7.5.4 Analytical analysis and comparison to experimental results . . . 125
7.6 Conclusions and recommendations . . . 126
8 Active Response Modifications of FWMAV-like Designs 129
8.1 Introduction . . . 130
8.2 Methods to induce structural property changes. . . 131
8.2.1 Piezoelectric polymers. . . 132
8.2.2 Embedded smart fluid. . . 134
8.2.3 Electrostatic softening. . . 136
8.3 Potential response modification . . . 137
8.3.1 Control using piezoelectric polymers . . . 139
8.3.2 Control using constrained ER-fluids . . . 142
8.3.3 Control using electrostatic softening. . . 145
8.3.4 Concluding remarks about different methods . . . 147
8.4 FWMAV maneuverability due to induced lift force difference. . . 147
8.5 Discussion . . . 150
8.6 Conclusions. . . 151
References. . . 152
9 Optimal FWMAV Wing Design 157 9.1 Introduction . . . 158
9.2 Methods . . . 159
9.2.1 Aerodynamic model . . . 159
9.2.2 Parameterized wing planform . . . 160
9.2.3 Wing flapping kinematics . . . 161
9.2.4 Overall wing performance . . . 162
9.2.5 Roll due to induced moment. . . 162
9.2.6 Combined formulation: wing performance and roll . . . 164
9.3 Results . . . 164
9.3.1 Wing design for energy-effective hovering flight . . . 164
9.3.2 Wing design for energy-effective hovering flight and roll control. . . 166
9.3.3 Kinematic control variables to achieve roll. . . 168
9.4 Conclusions. . . 171
References. . . 171
III Closure 175 10Discussion and Retrospection 177 10.1Applicability of framework for FWMAV designs. . . 178
10.1.1 Resonance response of FWMAV design . . . 179
10.1.2 Reflections regarding modeling . . . 179
10.2Experimental validation theoretical framework. . . 180
10.2.1 Initial validation resonance response modifications. . . 181
10.2.2 Difficulties for experiments at cm-scale . . . 181
10.3Methods to induce structural property changes. . . 181
10.3.1 Selection of methods to induce property changes . . . 182
10.3.2 Application of method in other context . . . 182
CONTENTS xi
11Conclusions and Recommendations 185
11.1Conclusions. . . 186
11.1.1 Active resonance response modifications . . . 186
11.1.2 Methods to induce structural changes. . . 188
11.1.3 Controllability of a resonant compliant FWMAV design . . . 189
11.2Recommendations . . . 191
11.2.1 Detailed model of the FWMAV design . . . 191
11.2.2 Control actuators . . . 192
11.2.3 The need for experimental validation . . . 192
A Experimental Validation: Resonance Response Modifications 195 A.1 Experimental setup and resonance responses. . . 196
A.2 Aim of the experimental validation . . . 196
A.3 Measured resonance response modification . . . 197
B Mimicked Hexbug: Control by Changing Leg Stiffness 201 B.1 Mimicked Hexbug design. . . 202
B.2 Control by changing the leg stiffness . . . 203
Acknowledgements 205
Curriculum Vitæ 207
S
UMMARY
This thesis studies a controllability approach for general resonant compliant systems. These systems exploit resonance to obtain a specific dynamic response at relatively low actuation power. This type of systems is often lightweight, is scalable and minimizes frictional losses through the use of compliant hinges. Some insect-inspired Flapping Wing Micro Air Vehicle (FWMAV) designs are based on such a resonant compliant sys-tem. These designs are carefully tuned such that a particular resonance response of the system corresponds to the desired wing flapping motion.
Most compliant systems do, however, require some form of control which aims at temporary modifications of the resonance response. For FWMAVs, for example, the nominal symmetric flap response needs to be modified into a temporary response con-taining an asymmetric component to enable maneuvering. The control is often com-plicated by stringent constraints on weight and power consumption. The system’s reso-nance response depends strongly on its structural properties (i.e., mass, damping, stiff-ness and their spatial distribution). Resonance response modifications can, thus, be controlled using carefully selected structural property changes.
PartIof this thesis presents a theoretical framework to determine the optimal loca-tion, number and dimensions of structural changes to obtain desired resonance modi-fications in the most power effective way. This framework provides a systematic design approach to modify: (i) undamped resonance modes, (ii) (non-)proportionally damped resonance modes and (iii) (non-)proportionally damped resonance responses. A pro-jection set is proposed to focus on the modifications at specific regions of interest, thus leaving the remaining portions of the response unspecified. This projection set allows the specification of a wide range of requirements (e.g., modify relative vibration ampli-tude or relative phase difference). Sensitivity analysis is used to linearly approximate the effect of structural changes on the resonance mode. These sensitivities are expressed in terms of the system’s modal basis to increase insight and intuitive understanding. This modal expansion revealed that resonance mode modifications are primarily determined by neighboring modes (i.e., modes for which the resonance frequency is close to the fre-quency of the resonance mode of interest). The framework requires, thus, only a limited number of dominant resonance solutions during analysis which reduces computational effort. Exploration of the modal basis as the preferred bases also showed that closely spaced resonance frequencies could potentially result in more effective control. Hence, considering the resonance frequency spacing while designing resonant compliant sys-tems might lead to more power effective control of these syssys-tems.
PartIIreviews methods to actually induce structural property changes within a sys-tem. To prevent a detailed analysis on all these methods, a qualitative comparison is performed using selection criteria that anticipate the application of these methods in dynamic, small-sized, lightweight systems, such as FWMAV designs. The used selec-tion criteria are, among other, response time, robustness, scaling and effectiveness. The
comparison resulted in four most promising candidates that rely on smart fluids, in-teracting stacked layers, electrostatic softening, or piezoelectric polymers. These four methods induce damping and stiffness changes. The effectiveness of a specific method to induce resonance response modifications depends, among other, on the structural property distribution of the reference system. For example, the control actuator designs in the present work which rely on either smart fluids or piezoelectric polymers do not introduce sufficient damping compared to the relatively high aerodynamic damping in FWMAV designs, to significantly modify the wing’s passive pitching motion. Effective FWMAV resonance response modifications due to structural property changes do, con-sequently, most likely rely on structural stiffness changes.
By sticking stacked layers (e.g., using electrostatics), relatively high absolute stiffness changes can be obtained. Analytical and experimental results show significant mod-ifications of the passive pitching motion of a FWMAV wing design if a control actua-tor design based on stacked layers is integrated in the wing design. The corresponding stroke-averaged lift force change was as much as 32%. This method introduces some inevitable frictional damping if the stacked layers slip with respect to each other. The long-term influence of this slip (e.g., wear) on the method’s performance is not studied. On the other hand, the theoretical change of the lift force is only limited to a few percent if a control actuator design based on electrostatic softening or piezoelectric polymers is integrated into the FWMAV wing design. However, for lightweight FWMAV designs a lift force difference of only a few percent might be sufficient to obtain the desired maneu-verability due to the low moment of inertia of the system. The flexibility of a method that relies on electrostatic softening is limited since the orientation of the electrodes is deter-mined in the design phase. Piezoelectric polymers form an appealing class of materials to achieve controllability of smart, lightweight and compliant FWMAV designs although experiments are needed to validate the practical applicability. To perform experimental validations there is a high demand for manufacturing tools and approaches for FWMAV-scale structures.
Energy-effective hovering and active flight control are of evident importance for the usefulness of FMWAVs. Chapt.9presents a combined approach to find an optimal wing design (i.e., wing planform and pitching kinematics) for energy-effective hovering and roll control. The most effective control variable to enforce the required body moment for a control action depends strongly on the wing design. The design of flapping wings requires, therefore, a combined approach to guarantee both energy-effective hovering and effective control.
This thesis contributes to the controllability of resonant compliant systems in gen-eral and the compliant FWMAV design in particular. Although there is yet no physical, controllable FWMAV design with integrated active components, the current research strongly indicates the applicability of structural property changes to reach controllability of this type of systems. Effective control requires an integrated approach that considers simultaneously the structure, the wings, the kinematics, the methods to induce property changes and the desired controllability.
S
AMENVAT TING
Dit proefschrift onderzoekt een aanpak voor de regeling van resonante compliante sys-temen. Deze systemen maken gebruik van resonantie om energiezuinig een specifieke dynamische responsie te verkrijgen. Dit type systeem is extreem licht, schaalbaar en heeft minimale wrijvingsverliezen door het gebruik van compliante scharnieren. Som-mige insect-geïnspireerde micro vliegmachines met flapperende vleugels, zogenaamde “Flapping Wing Micro Air Vehicles (FWMAVs)”, maken gebruik van deze resonante com-pliante systemen. Het ontwerp van deze FWMAVs is zorgvuldig getuned zodat een spe-cifieke resonantieresponsie overeen komt met de gewenste vleugel flapbeweging.
Compliante systemen vereisen vaak een bepaalde regeling om een tijdelijke veran-dering van de resonantieresponsie te realiseren. Voor FWMAVs betekent dit bijvoor-beeld dat een symmetrische flap responsie tijdelijk verandert wordt in een responsie met asymmetrische componenten wat maneuvers mogelijk maakt. De regeling wordt vaak bemoeilijkt door restricties in gewicht en energieverbruik. De resonantieresponsie is af-hankelijk van de structurele eigenschappen van het systeem (oftewel de verdeling van massa, demping en stijfheid in het systeem). Deze responsie kan dus verandert worden door zorgvuldig gekozen aanpassingen van deze eigenschappen.
DeelIvan dit proefschrift presenteert een theoretisch raamwerk om de locatie, het aantal en de grootte van de structurele aanpassingen te bepalen die resulteren in de meest energie-efficiënte, gewenste verandering van de resonantieresponsie. Het raam-werk geeft een systematische aanpak voor de regeling van: (i) ongedempte resonantie-modes, (ii) (niet-) proportioneel gedempte modes en (iii) (niet-) proportioneel gedempte resonantieresponsies. Een projectie-set is geïntroduceerd om enkel te focussen op de verandering van een specifiek gedeelte van de resonantiemode zonder naar de rest te kijken. Met deze projectie kunnen veel eisen gesteld worden aan de beoogde verande-ring (bijvoorbeeld, relatieve amplitude verandeverande-ringen van de FWMAV vleugel flapbewe-ging). Het effect van structurele aanpassingen op de resonantiemodes is benadert met behulp van een gevoeligheidsanalyse. Om meer intuïtief inzicht te verkrijgen is deze ge-voeligheidsanalyse uitgedrukt in de modale basis van het systeem. Deze modale expan-sie laat zien dat de verandering van een resonantiemode voornamelijk bepaald wordt door de dichtbij gelegen modes (oftewel modes waarvan de resonantiefrequentie dicht-bij de frequentie ligt van een mode die verandert moet worden). Een analyse vereist dus maar een beperkt aantal dominante modes wat zorgt voor een verminderde rekentijd. Het uitdrukken in de modale basis laat tevens zien dat de regeling effectiever kan wor-den door dicht bij elkaar gelegen resonantiefrequenties. Voor een effectieve regeling van resonante compliante systemen moet de afstand tussen de resonantiefrequenties dus meegenomen worden tijdens het ontwerpproces.
DeelIIgeeft een overzicht van methoden om structurele aanpassingen te realiseren in een systeem. Om geen gedetailleerde analyse van al deze methodes te hoeven doen is een kwalitatieve vergelijking uitgevoerd op basis van criteria die inspelen op de
baarheid van deze methodes in een dynamisch, relatief klein, lichtgewicht systeem zoals een FWMAV ontwerp. Deze selectiecriteria zijn, onder andere, reactietijd, robuustheid, schaalbaarheid en effectiviteit. De vergelijking resulteerde in vier veelbelovende kandi-daten: actieve vloeistoffen, interacterende gestapelde lagen, elektrostatisch verslappen en piëzo-elektrische polymeren. Deze methodes induceren dempings- en stijfheidsaan-passingen. De effectiviteit van een bepaalde methode om responsieveranderingen te realiseren hangt af van de structurele eigenschappen van het referentiesysteem. De ac-tuatoren in dit werk die gebaseerd zijn op actieve vloeistoffen of piëzo-elektrische po-lymeren induceren bijvoorbeeld onvoldoende demping in vergelijking met de relatief hoge aerodynamische demping in FWMAVs, om significante veranderingen in de pas-sieve invalshoek van de flapperende vleugel te realiseren. Effectieve veranderingen van de FWMAV-responsie komen daardoor waarschijnlijk van stijfheidsaanpassingen.
De absolute stijfheidsaanpassing is relatief hoog wanneer gestapelde lagen met be-hulp van bijvoorbeeld elektrostatica tijdelijk op elkaar geplakt worden. Analytische en experimentele resultaten laten significante veranderingen van de passieve invalshoek van een FWMAV vleugel zien, wanneer een actuator gebaseerd op gestapelde lagen in deze vleugel is geïntegreerd. De overeenkomstige verandering van de gemiddelde lift-kracht is 32%. Wanneer de gestapelde lagen niet plakken maar over elkaar heen glijden zorgt dat voor onvermijdelijke demping door wrijving. De langetermijn invloed van deze wrijving (bijvoorbeeld slijtage) op het functioneren van deze methode is niet bestudeerd. De theoretische verandering van de liftkracht is maar een paar procent wanneer een ac-tuator gebaseerd op elektrostatische verslapping of piëzo-elektrische polymeren in het vleugel ontwerp is geïntegreerd. Echter, een liftkracht verschil van enkele procenten is mogelijk voldoende om de gewenste maneuvreerbaarheid te behalen van lichtgewicht FWMAVs met een laag traagheidsmoment. De methode die gebaseerd is op elektrosta-tische verslapping heeft weinig flexibiliteit omdat de oriëntatie van de elektrodes al in de ontwerpfase vastgesteld wordt. Piëzo-elektrische polymeren zijn aantrekkelijk om de regeling van elegante, lichtgewichte en compliante FWMAV ontwerpen te realiseren hoewel experimenten nodig zijn om de praktische toepasbaarheid te onderzoeken. Er is een sterke behoefte aan fabricage- en assemblagemiddelen om experimentele validaties uit te voeren van systemen op de FWMAV-schaal.
Voor het vliegen met en de besturing van FWMAVs is energie-efficiëntie van evident belang. Hst.9presenteert een gecombineerde aanpak om het optimale vleugel ontwerp te vinden (oftewel vleugel lay-out en flap kinematica) voor energie-efficiënt vliegen en effectieve besturing. De meest effectieve besturingsvariabele om het vereiste lichaams-moment te genereren voor een bepaalde maneuver hangt sterk af van het vleugelont-werp. Het ontwerp van flapperende vleugels vereist dus een gecombineerde aanpak om zowel energie-efficiënt vliegen als een effectieve besturing te garanderen.
Dit proefschrift bevordert de regeling van resonante compliante systemen in het al-gemeen en het FWMAV ontwerp in het bijzonder. Hoewel er op dit moment nog geen fysiek, bestuurbaar FWMAV ontwerp is met geïntegreerde actieve componenten, laat dit onderzoek duidelijk zien dat de regeling van dit type systemen haalbaar is met nauw-keurig gekozen structurele aanpassingen. Effectieve regeling vereist een geïntegreerde aanpak waarbij het structurele ontwerp, de vleugel kinematica, de methode om de struc-turele aanpassing te doen en de gewenste regeling tegelijkertijd bekeken worden.
1
I
NTRODUCTION
1
1.1.
B
ACKGROUND TO
FWMAV
DESIGN
Scientists and engineers show a passionate perseverance in the study and development of Micro Air Vehicles (MAVs). The United States Defense Advanced Research Projects Agency (DARPA) defined the size of a MAV to be limited to 150 mm in length, width and height. Research focuses either on downscaling traditional fixed and rotary wing aircrafts or on applying biologically inspired flapping wing configurations. The latter is appealing since, for example, flying insects demonstrate an astonishing speed and agility during daily life flight which would be desirable features for MAV designs [1].
MAVs are applied in surveillance (e.g., police, security, and military), inspection of in-accessible or dangerous locations (e.g., disaster scenes, sewers, and pipes) and in sens-ing and mappsens-ing of 3D spaces (e.g., particle concentrations, temperature, and radia-tion). MAVs can even be deployed to autonomously pollinate a field of crops in the case of a decrease in bee populations [2]. For indoor applications, fixed wing MAVs are less applicable since they require sufficient forward flight to stay aloft. Rotary wing and flap-ping wing MAVs can, on the other hand, hover in confined, small spaces. Since flapflap-ping wing MAVs exploit unconventional force generation mechanisms which correspond to unsteady flows, they outperform fixed wing and rotary wing MAVs in terms of aerody-namic efficiency in the intended flight regimes [3].
For lightweight flapping wing MAV designs biology is appealing as a source of inspi-ration to decrease both weight and power consumption, which are of paramount impor-tance. To minimize power demands, many insects appear to use the resonant properties of their entire thorax-wing structure to accelerate and decelerate their wings during flap-ping flight [4,5]. At resonance, the system (i.e., the thorax-wing structure in the example of insects) stores and transfers energy most effectively between different storage modes (i.e., kinetic energy and elastic energy). The only energy to be compensated for by the input excitation is, ideally, determined by the energy dissipated by damping.
One way to improve the performance on MAVs weight and power consumption is, subsequently, through the use of a resonant compliant mechanism to drive the flapping motion of the wings. This could result in highly integrated, insect-inspired and compli-ant structures of minimum mass that
1. have a reduced energy signature due to the advantage of exploiting energy storing mechanisms;
2. minimize losses through the use of compliant hinges; and 3. maintain the possibility for further downscaling.
Previous research within the Atalanta project resulted, conform the above mentioned strategy, in the insect-inspired design and realization of a compliant four wing FWMAV design that exploits its resonance properties to decrease power consumption [6] (see Fig.1.1a). The Atalanta project aims to develop a robust Flapping Wing Micro Air Vehi-cle (FWMAV) with an intended dimension of 100 mm wingspan and a maximal weight of 4 gram. Eventually, the FWMAV should be able to: (i) hover and fly autonomously; (ii) communicate effectively within a swarm and with the operator; (iii) manage power storage; and (iv) provide payload capacity (e.g., to carry sensors). Consequently, the
Ata-1.2.NECESSITY FOR CONTROL
1
3(a)Realization of FWMAV design with the actu-ator within the ring-type mechanism, from [6].
spring
actuator mass
mass, damping (wing)
(b)Schematic of the FWMAV design indicating the connection between the structural parts. Figure 1.1: Realization and schematic representation of the four wing compliant FWMAV design.
lanta project is multi-disciplinary with scientific challenges in mechanics, electronics, aerodynamics, control and informatics.
The FWMAV design of Fig.1.1aconsists of a ring-type compliant mechanism that converts the linear harmonic actuation into a repetitive flapping motion of the wings via a planar four-bar mechanism. Proper tuning of all components of this design (i.e., mass, damping, stiffness and their spatial distribution) leads to the desired flapping motion if the structure is excited in a particular resonance frequency. Although the structure of Fig.1.1ahas multiple resonance frequencies, the excitation frequency that leads to the desired resonance flapping motion, or resonance response, is close to 27 Hz [7]. The damping of this system is primarily determined by the aerodynamic damping around the wings. Fig.1.1bshows a schematic representation of the compliant FWMAV design indicating the interconnection between the lumped masses, the wings with mass and damping, the springs and the actuator within the ring-type mechanism.
1.2.
N
ECESSITY FOR CONTROL
Although the basic wing motion needed for flapping flight can be effectively achieved with the FWMAV design of Fig.1.1a, practical use of this design requires flight control. This is needed for stabilizing basic flight, hovering, maneuvering, and flying mode tran-sitions (e.g., from hovering to forward flight). The traditional aviation approach would add components dedicated to control (e.g., the tail of an airplane), or use advanced com-ponents to change the structural geometry (e.g., ailerons and flaps in aircraft wings). Examples of this approach in FWMAV designs are the DelFly [8], and the Caltech Mi-crobat [9]. This approach is, however, difficult to miniaturize and it requires additional components and actuators which results in increases in weight and power consumption. Additionally, it lacks clear biological similarities.
The inspiration from biology (e.g., insects) has been hindered by difficulties in dis-tinguishing the complex dynamic effect of control muscles responsible for their agile behavior. Dragonflies are, for example, known to be extremely versatile fliers and they appear to have a unique degree of control over the flapping muscles of the individual wings. They show takeoff accelerations of up to 2g with maximum flight speeds up to 10 m/s [10] and turning accelerations exceeding 4g [11]. Additionally, dragonflies at
rel-1
atively low speeds can perform a 90general, power muscles to obtain the main flapping motion and a great number of con-◦ yaw turn in two wing beats [12]. Insects use, in trol muscles (i.e., for some insects even 22 pairs that account for only < 3% of the flight muscles mass) to fine-tune this motion [13,14]. Experimental studies have revealed that the fruit fly employs a simple bias of its wing hinge to create a control torque [15]. Only recently, time-resolved microtomography has been utilized to visualize the three-dimensional movements of the five largest control muscles of blowflies [16]. This tech-nique revealed rather complex, unexpected non-linear behavior and buckling of parts of the structure during flight control.As inspired by the elegant, integrated biological solutions for control, several at-tempts have been made to integrate small-scale control mechanisms within FWMAV designs. An example of such an approach is demonstrated on the Harvard Microrobotic Fly (HMF). Recent work used aerodynamic dampers to stabilize the FWMAV around its roll and pitch axes [17], altered the transmission ratio of the left and right wing stroke amplitude to produce appreciable control torques [18], gave each wing its own control actuator [19], and created control torques by an indirect modulation of the wings’ angle of attack [20]. The research efforts resulted in a controlled hovering of this insect-size design (i.e., 30 mm wingspan and 60 mg weight) [21]. An attempt to resolve the adverse high level of mechanical complexity that arise from this approach utilizes microelec-tromechanical systems (MEMS) techniques [22].
Despite all research efforts to control FWMAV designs, an energy efficient, highly integrated (as demonstrated by insects), elegant and low weight solution to achieve con-trollability of the resonance flap response of compliant FWMAV designs does not exist.
1.3.
C
ONTROLLABILITY OF RESONANT COMPLIANT
FWMAV
S
The approach to achieve controllability of resonant compliant FWMAV designs needs to be integrable and energy efficient with a minimum of auxiliary structures and weight to maintain the basic advantages of these systems. The approach needs, additionally, to be scalable to enable future downscaling. Potential approaches are listed below.
Approach A : divide the initial resonant system into several (delicately) coupled res-onators which allow for a difference in actuation (i.e., amplitude and phase) be-tween these resonators. For the FWMAV design this would imply that all individ-ual wings can, for example, be connected to separate resonators to enable control of the flapping motion of the individual wings (see Fig.1.2a).
Approach B : proper tuning of the mass and stiffness distribution of the basic resonant system to introduce different resonance responses corresponding to different driv-ing frequencies (see Fig.1.2b). For the FWMAV design this would imply that the different resonance responses are, for example, chosen such that the flapping wing motion of the different responses correspond to different flight configurations. Changing from one response to another only requires a change in driving fre-quency using the basic actuator.
Approach C : induce structural property changes (i.e., changes in mass, damping, stiff-ness or their spatial distribution) to parts of the basic resonant system to modify
1.3.CONTROLLABILITY OF RESONANT COMPLIANTFWMAVS
1
5separate resonator
(a)Approach A: divide basic
res-onant system in separate, indi-vidually controllable resonators.
tuned
stiffness tuned mass
(b) Approach B: well-tuned
property distribution resulting in multiple resonance responses.
variable stiffness
(c) Approach C: induce
struc-tural property changes to modify the resonance response. Figure 1.2: Schematics of the different approaches to control resonant compliant FWMAV designs.
the resonance response (see Fig.1.2c). For the FWMAV design this would, for ex-ample, imply that the flapping motion of one of the wings is modified due to a carefully induced structural property change.
The aforementioned approaches to achieve FWMAV controllability aim at obtaining the desired resonance response modification while still exploiting resonance. Each of these approaches has its own challenges.
Approach A requires several excitation actuators. Although each one can be smaller, the added overhead (i.e., wiring, power management and auxiliary structures) will in-crease the total mass and system’s complexity. This approach considers the system in separate parts instead of considering it as a whole which complicates further downscal-ing. The efficiency and applicability of the method depends largely on the actuator’s de-sign (e.g., its power density) and on the possibility to tune the influence of the separate resonators on the total system. Approach B requires a complicated resonator design for sufficient variation between the resonance responses to allow for control. Because some parts are not driven at resonance, there will be additional losses in the system. Addi-tionally, the required power to change from one resonance response to an other might be significant. During control, the driving frequency is deliberately changed to excite the desired resonance response. However, during the gradual change of the driving fre-quency (i.e., it can not be changed instantly since the system needs time to adapt), unde-sired resonance modes might be excited with obvious, adverse consequences for control. Approach C requires additional actuators or smart materials (e.g., piezoelectric material or electroactive polymers) to change the intrinsic properties of structural components. These actuators do not need to provide the power necessary to create lift, but only to change the structural properties sufficiently to allow for the desired resonance response modifications.
Based on the preceding arguments, Approach C appears most promising to control lightweight, resonant compliant systems in an elegant, effective and integrated manner. Fig.1.3demonstrates the conceptual idea of this approach on a simplified FWMAV-like structure. Fig.1.3ashows a symmetric design consisting of a ring structure with four cantilever-like wings attached to it. The structure is harmonically excited by the actuator inside the ring. The resonance response due to this excitation is symmetric and shown in Fig.1.3b. Hence, all wings flap with the same amplitude. Fig.1.3cindicates that the
1
structural properties (e.g., the bending stiffness) of part of the ring can be varied usingany suitable actuator. Due to the induced structural property change, the resonance response is modified (i.e., the ring deforms differently) as shown in Fig.1.3d. Conse-quently, the flapping amplitude of all wings is no longer the same. This modification of the resonance flap response demonstrates the conceptual idea of this controllability approach.wing
ring
actuator
(a)Simplified, perfectly symmetric compliant FWMAV design with four cantilever-like wings attached to a ring structure. The actuator in-duces a harmonic deformation of the ring.
φ1 φ2
φ1= φ2
(b)Symmetric flapping of the wings around the reference configuration due to the excitation, represented by the red arrow. The flapping am-plitude of all wings is the same (i.e.,φ1= φ2).
variable structural
property
(c) Location on the ring structure at which structural properties (e.g., bending stiffness) can be varied by any suitable actuator.
φ1 φ2
φ1> φ2
(d)Asymmetric flapping due to the excitation as a result of the introduced property change (i.e.,φ1> φ2). The effect is exaggerated for
clar-ity reasons.
Figure 1.3: Sketches of a simplified perfectly symmetric, compliant FWMAV design for which the symmetric resonance response of the four cantilever-like wings due to a harmonic excitation becomes asymmetric after inducing structural property changes to the ring structure.
1.4.THESIS OBJECTIVE
1
71.4.
T
HESIS OBJECTIVE
This thesis focuses on the use of (local) structural property changes to achieve control-lability of resonant compliant systems (e.g., the FWMAV design). To obtain the desired resonance response modification the required location and magnitude of the structural property change must be determined. The emerging and more abstract fundamental question considers, subsequently, the determination of property changes that result in the most effective, desired response modifications of general resonant compliant sys-tems. These structural property changes have to be induced by small actuators. To this end, actuators are required that are lightweight, integrable and consume minimal energy while allowing for sufficient variation of the resonance response to enable control. Addi-tional points of concern are overhead minimization (such as minimal wires and minimal support structure) and required downscaling possibilities.
The general research question is:
What structural property changes can achieve controllability of resonant compliant systems within the scope of lightweight, low power, scalable and simple Flapping Wing Micro Air Vehicle designs?
In addition to the control of FWMAV designs, the use of resonant compliant systems is interesting for other applications as well (e.g., MEMS devices or energy harvesters) for reasons of robustness, low wear, and energy efficiency. In many fields, power require-ments are becoming more stringent and systems will need to become lighter. Conse-quently, it is likely that more systems will make use of fully compliant resonant config-urations to achieve this. Proper handles are needed to achieve controllability of these systems without giving up on the advantages associated to resonant compliant systems. The results will, additionally, be of interest for a variety of systems that are influenced by resonance. Traditionally, these systems apply discrete mass, dampers and springs to isolate the vibration to parts of a structure. However, by modifying the systems dynamic behavior with integrated active components that induce structural property changes, it would be possible to achieve better isolation with less bulky systems. For example, car engine mountings are traditionally tuned to optimally damp the vibration at a cer-tain rotational speed. However, by actively changing the stiffness of this mounting, the damping could be optimized to maximize damping for all possible rotational speeds. As another example, the mechanical properties of (compliant) MEMS devices (e.g., sensors) are often difficult to define accurately during fabrication. Integrated active components could apply property changes to optimize performance.
1.5.
T
HESIS OUTLINE
This thesis consists of eight research chapters, two concluding chapters and two appen-dices. The eight research chapters are divided into two parts. PartIis solely theoretical, while PartIIalso contains some experiments. The eight research chapters are primarily based on conference and journal papers, which causes some overlap between the chap-ters. Fig.1.4shows a graphical overview of the thesis outline.
PartI: Actively modifying resonance responses PartIapplies a rather general and fun-damental approach to study to which extent resonance modes and resonance responses
1
can be modified by actively induced (local) structural property changes. Note the cleardistinction between a resonance mode and a resonance response:A resonance mode is defined as one of the system’s mode shapes corresponding to
a specific resonance frequency (or eigenvalue). A resonance mode corresponds to one of the free-vibration resonance solutions (or eigensolutions) of a system.
A resonance response is defined as the system’s dynamic response whenever
ex-cited in one of its resonance frequencies. A typical example is the resonance flap-ping motion of the FMWAV when excited in one of its resonance frequencies. The resonance response of a system excited in one of its resonance frequencies will be primarily determined by the resonance mode corresponding to that frequency [23], but is not identical due to the presence of other resonance modes.
Chapt.2demonstrates that (local) structural property changes can result in signif-icant resonance mode modifications and, therefore, in the controllability of resonance modes. Chapt.3presents, subsequently, a general framework to determine the location at which structural property changes lead to the most effective undamped resonance mode modifications. This effectiveness depends on the resonance frequency spacing (i.e., difference between consecutive resonance frequencies). Chapt.4extends the frame-work of Chapt.3by investigating the influence of (non-proportional) damping on deter-mining the location at which property changes are most effective in modifying damped resonance modes.
After studying resonance modes or mode shapes in Chapts.2–4, Chapt.5focuses on damped resonance responses. It presents a modal-based approach to determine the lo-cation at which structural property changes (or excitation adjustments) lead to the most effective (non-proportionally) damped resonance response modifications. This effec-tiveness is, again, influenced by the system’s resonance frequency spacing but also by the amount of damping. All chapters in PartIuse general systems (i.e., mass-(damper)-spring systems and plates) to demonstrate the theoretical framework.
PartII: Inducing structural property changes PartIIdiscusses methods and actua-tion principles to actually apply the structural property changes which were implicitly assumed to be realizable in PartI. Chapt.6provides an overview of different methods to induce structural property changes. A qualitative criteria-based comparison is per-formed to identify the most applicable candidates to achieve controllability of dynamic, compliant and lightweight systems like the FWMAV design. These comparison criteria are, for example, response time, robustness, and power efficiency. Chapt.7and8study the resulting most promising candidates more thoroughly both theoretically and exper-imentally. Both chapters investigate the influence of the structural property changes on the FWMAV flap response. As a first attempt, we focused on modifying the flapping mo-tion of a single, isolated flapping wing instead of considering the complete four wing FWMAV design. Since the FWMAV is a highly integrated and compliant system, modifi-cations of the flapping motion of a single wing will, however, result in changes of the total resonance response. Chapt.9presents a study to determine FWMAV wing designs that maximize the average aerodynamic lift force modifications due to control parameters such as structural property changes.
1.5.THESIS OUTLINE
1
9PartIII: Closure Chapt.10discusses the implications of the current work for the con-trollability of resonant compliant systems in general and the FWMAV design in partic-ular. Although there is yet no physical, controllable resonant compliant FWMAV design with integrated active components, the current research strongly indicates the appli-cability of structural property changes to reach controllability of this type of FWMAV designs. Chapt.11presents the conclusions from the research in the different parts. Ad-ditionally, directions of further research are indicated.
Finally, two appendices are added. Appx.Apresents an initial experimental valida-tion of resonance response modificavalida-tions. This validavalida-tion uses a clamped-free cantilever beam. Appx.Bdemonstrates controlled steering of a dynamically complex, walking plat-form (i.e., the mimicked Hexbug), by actively changing the leg’s bending stiffness.
Appx.A: Initial experimental validation Appx.B: Control of mimicked Hexbug Ch.10: Retrospection
and Discussion
Ch.11: Conclusions and Recommendations
PartIII: Closure
Ch.6: Overview of methods to induce structural changes Ch.7: Electrostatic sticking to modify flap kinematics Ch.8: in-depth study of 3 ways to modify flap kinematics Ch.9: wing designs that improve control parameter sensitivity
PartII: Inducing structural property changes
Ch.2: Applicability of local changes
to modify
resonance modes Ch.3: General framework to modify undamped resonance modes Ch.4: General framework to modify damped resonance modes Ch.5: General framework to modify damped resonance responses
PartI: Actively modifying resonance responses Ch.1: Introduction
1
R
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[5] T. Weis-Fogh, Energetics of hovering flight in hummingbirds and in drosophila,The Journal of Experimental Biology 56, 79 (1972).
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insect-inspired design,Ph.D. thesis, Delft University of Technology (2010).
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actuation mechanism for application in flapping wing mavs,International Journal of Micro Air Vehicles 1, 263 (2009).
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Design, aerodynamics, and vision-based control of the delfly,International Journal of Micro Air Vehicles 1, 71 (2009).
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envelope and its application to micro uav research and development, inIntelligent Autonomous Vehicles, Vol. 8 (2013) pp. 231–234.
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mus-cles of the blowfly (calliphora vicina),Journal of Comparative Physiology A: Sensory, Neural, and Behavioral Physiology 178, 813 (1996).
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bio-logically inspired, insect-scale robot,Science 340, 603 (2013).
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[23] M. Géradin and D. J. Rixen,Mechanical Vibrations: Theory and Application to Struc-tural Dynamics, 3rd ed. (John Wiley & Sons, 2014).
I
A
CTIVELY
M
ODIFYING
R
ESONANCE
R
ESPONSES
2
E
IGENSOLUTION
M
ODIFICATIONS
DUE TO
L
OCAL
S
TRUCTURAL
P
ROPERTY
C
HANGES
This thesis proposes to use structural property changes to achieve controllability of res-onant compliant systems. However, this requires that the system’s eigensolutions (i.e., the eigenvalues (or resonance frequencies) and corresponding mode shapes (or resonance modes)) do actually modify in a predictable and effective manner as a function of these property changes. This chapter studies the eigensolution evolution of undamped and damped mechanical systems when some system property is changed over several orders of magnitude (i.e., this chapter studies the system’s modal behavior as a function of some system property). Three distinct phases are identified in the eigensolution evolution which have significant implications for the design, optimization and control of dynamic systems. This chapter is organized as follows. Sec.2.1introduces several characteristic modal phe-nomena that might occur while studying the eigensolution evolution as a function of property changes. After that, Secs.2.2and2.3study the eigensolution evolution of un-damped and un-damped discrete systems as a function of relatively large local stiffness and damping changes, respectively. Sec.2.4reviews practical applications of modal phenom-ena from literature. Some practical implications for the design and control of dynamic systems are briefly discussed in Sec.2.5. Finally, Sec.2.6concludes that property changes are applicable to control the eigensolutions of resonant compliant systems.
2
2.1.
I
NTRODUCTION
For the design, optimization and control of dynamic systems, modal parameters are of-ten used to characterize the system behavior, see, for example, [1]. This requires the calculation of eigensolutions (i.e., the eigenvalues and mode shapes). Mode shapes can be both global (i.e., involving the entire system) or local (i.e., confined to a specific part of the system or subsystem). Additionally, the eigensolution modification due to (local) system property changes is often required (e.g., in optimization). The eigensolutions might modify either drastically or hardly due to these changes.
This study on the eigensolution evolution as a function of (local) structural property changes revealed interesting characteristic modal behavior. We induced a wide range of local structural stiffness and damping changes to discrete systems and observed the eigensolution evolution and corresponding modal phenomena as a function of these changes. For this range with relatively large local property changes (i.e., if s represents a property variable, then |∆s|/|s| ¿ 1 does not hold), the eigensolution evolution demon-strates three distinct phases which have great practical implications. In these phases, modal phenomena like mode shape localization, modal coupling, eigenvalue veering, and eigenvalue crossing, can be recognized.
Although our study focuses on (local) structural stiffness and damping changes in mechanical systems, these modal phenomena are also recognized in other fields. In solid-state physics, Anderson localization predicts that an electron may become immo-bile (i.e., eigenstate localization) when placed in a disordered lattice [2]. In structural mechanics, curve veering was recognized in vibrating rectangular plates when eigenval-ues were plotted against the ratio between the length and width of the plate [3]. More recently, the research focused on the quantification [4] and experimental validation [5] of these modal phenomena.
Despite the received attention, the practical implications of these modal phenomena during the design, optimization and control of dynamic systems are not well established in literature. As a first step, we study in Secs.2.2and2.3the characteristic modal phe-nomena of undamped and damped discrete systems as a function of relatively large local stiffness and damping changes, respectively. Although we show small discrete mechan-ical systems (i.e., a symmetric 6-degrees of freedom (DOF) system), equivalent modal phenomena are observed for large scale, complex dynamic systems.
Sec.2.4reviews practical applications of the characteristic eigensolution evolution as a function of structural property changes from literature (e.g., local mode shapes and mode veering) to raise the awareness of these modal phenomena. Sec.2.5discusses briefly some corresponding practical implications of these modal phenomena for the design, optimization and control of dynamic systems. Finally, Sec.2.6concludes that structural property changes are applicable to achieve significant eigensolution modifi-cations and, thus, controllability, of resonance frequencies and resonance modes.
2.2.
E
IGENSOLUTION EVOLUTION DUE TO LOCAL STRUCTURAL
STIFFNESS CHANGES
This section studies the eigensolution evolution of an undamped mechanical system as a function of a range of local stiffness changes. It should be noted that similar modal
phe-2.2.LOCAL STRUCTURAL STIFFNESS CHANGES
2
17 m m m m m m kβ k k k kβ q1 q2 q3 q4 q5 q6Figure 2.1: Unconstrained 6-DOF mechanical system with local, variable spring stiffness indicated by the red arrows.
nomena would be found for local changes to the mass. Fig.2.1shows an unconstrained 6-DOF mechanical system with uniform mass denoted by m, and a non-uniform, vari-able stiffness distribution denoted by k for the center springs and by kβ = βk for the outside springs. The parameterβ determines the relative local stiffness which is always positive (i.e.,β > 0). The location at which local stiffness changes are induced implicitly implies localized modal behavior to happen as a function ofβ. We study longitudinal free vibrations, denoted by qnfor n = 1,...,6. The corresponding linearized equations of motion can be written as
mI∂ 2q ∂t2+ kAβ
£
β¤q = 0, (2.1)
where I ∈ R6×6indicates the identity matrix, q ∈ R6denotes the set of displacements, t represents time, and Aβ£
β¤ ∈ R6×6specifies the springs for which the stiffness is scaled
withβ. That is,
Aβ£ β¤ = β −β 0 0 0 0 −β β + 1 −1 0 0 0 0 −1 2 −1 0 0 0 0 −1 2 −1 0 0 0 0 −1 β + 1 −β 0 0 0 0 −β β . (2.2)
By introducing, successively, the normalization ¯v = q/l where l indicates a chosen length scale, the reference angular frequencyω0=pk/m, and the dimensionless timeτ = ω0t ,
Eq. (2.1) can be normalized as I∂
2¯v ∂τ2+ Aβ
£
β¤ ¯v = 0. (2.3)
If we assume a solution in the form of ¯v = vsin(ατ) and substitute this into Eq. (2.3), the eigenproblem to determine the real eigenvalues α2n ∈ R and corresponding mode shapes vn∈ R6as a function ofβ, can be given by
¡−α2
nI + Aβ £
β¤¢vn= 0 for n = 1, . . . , 6. (2.4) The structural eigenvalues corresponding to the mode shapes vn are, with the intro-duced dimensionless timeτ, denoted by ωn= αnω0. For allβ, the first eigensolution of this system is a rigid body solution (i.e.,α21= 0 ∀ β). Fig.2.2shows the evolution of the square root of the eigenvaluesα2jfor j = 2,...,6 as a function of β (i.e., αj£
2
10-2 100 β 102 β ¿ 1 β ∼ 1 β À 1 10-1 100 101 αj α2 α3 α4 α5 α6Figure 2.2: Evolution of the square root of the 2ndto 6steigenvalueα2jon a bi-logarithmic scale as a function ofβ. These eigenvalues determine, with the reference angular velocity ω0, the non-zero structural eigenvalues
of the 6-DOF system (i.e.,ωj= αjω0).
From Fig.2.2, we distinguish three phases:
1. β ¿ 1, for which α2andα3approach zero although they do not become rigid body solutions (i.e.,α1is the only rigid body solution) andα4,α5, andα6are relatively insensitive;
2. β ∼ 1, for which the eigenvalues converge towards each other although they do not cross but veer away (i.e., eigenvalue veering phenomena). In this phase, all eigenvaluesαjmodify relatively significant as a function ofβ; and
3. β À 1, for which α2,α3, andα4are (almost) insensitive whileα5andα6veer away with one another and become (nearly) repeated while increasing in a straight line withβ.
Consequently, all eigenvaluesαjdo exhibit relatively sensitive and insensitive phases as a function ofβ.
Fig.2.3shows the mode shapes v2and v5at three values ofβ corresponding to the
distinguished phases (i.e.,β ¿ 1, β = 1, and β À 1). The mode shapes are normalized as vTjIvj= 1. The longitudinal vibration is represented vertically for visualization reasons. For the three distinct phases of the mode shape evolution as a function ofβ we observe:
1. β ¿ 1, for which all mode shapes vj are relatively insensitive as a function ofβ. For modes v2and v3, the deformation is highest in the relatively compliant springs
with stiffness kβ(e.g., mode shape v2is localized, see Fig.2.3) which explains that α2andα3approach zero ifβ goes to zero.
For modes v4, v5and v6, the deformation is highest in the springs with constant
stiffness k. Consequently, the corresponding eigenvalues (i.e.,α4,α5, andα6) do hardly modify as a function ofβ;
2.3.LOCAL STRUCTURAL DAMPING CHANGES
2
19 DOF 1 6 -1 1 DOF 1 6 -1 1 DOF 1 6 -1 1 DOF 1 6 -1 1 DOF 1 6 -1 1 DOF 1 6 -1 1 v2 v5 β ¿ 1 β = 1 β À 1Figure 2.3: Mode shapes v2and v5of the 6-DOF mechanical system for the three different phases (the red
arrows indicate the location at which relative stiffness changes are applied).
2. β ∼ 1, for which all mode shapes vjare global and well known from systems with (almost) uniform mass and stiffness properties. For this phase, the mode shapes are relatively most sensitive as a function ofβ; and
3. β À 1, for which all mode shapes vj are relatively insensitive as a function ofβ. For modes v2, v3and v4, the deformation is highest in the springs with constant
stiffness k. Hence,α2,α3, andα4do hardly change as a function ofβ. The springs with stiffness kβact as rigid part. For modes v2and v3, the location of highest
deformation shifts compared to Phase 1 (see v2in Fig.2.3forβ ¿ 1 and β À 1).
For modes v5and v6, the deformation is highest in the relatively stiff springs with
stiffness kβ(i.e., localized mode shapes) which explains thatα5andα6continue to increase whenβ increases.
Consequently, for relatively large local stiffness differences, localized mode shapes ap-pear (i.e., the system apap-pears to divide into subsystems). The mode shapes are most sensitive to changes inβ for (almost) uniform properties.
2.3.
E
IGENSOLUTION EVOLUTION DUE TO LOCAL STRUCTURAL
DAMPING CHANGES
This section studies the eigensolution evolution of a mechanical system as a function of a range of local damping changes. Fig.2.4shows an unconstrained 6-DOF mechanical system with uniform mass denoted by m, uniform stiffness denoted by k, and a local variable viscous damping denoted by cη= η c. The parameter η determines the local vis-cous damping which is always positive (i.e.,η > 0). The positions of the local dampers implicitly implies localized modal behavior to happen as a function ofη. We study
lon-2
m cη m m m m m k k k k k cη q1 q2 q3 q4 q5 q6Figure 2.4: Unconstrained 6-DOF mechanical system with local, variable viscous damping indicated by the red arrows.
gitudinal free vibrations, denoted by qn for n = 1,...,6. The corresponding linearized equations of motion can be written as
mI∂ 2q ∂t2+ cAη £ η¤∂q ∂t+ kAkq = 0, (2.5)
where I ∈ R6×6indicates the identity matrix, q ∈ C6denotes the complex set of displace-ments, t represents time, Aη£
η¤ ∈ R6×6specifies the local dampers for which the
damp-ing is scaled withη, and Ak∈ R6×6specifies the stiffness distribution. That is,
Aη£ η¤ = η −η 0 0 0 0 −η η 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 η −η 0 0 0 0 −η η and Ak= 1 −1 0 0 0 0 −1 2 −1 0 0 0 0 −1 2 −1 0 0 0 0 −1 2 −1 0 0 0 0 −1 2 −1 0 0 0 0 −1 1 . (2.6)
By introducing, similar to Sec.2.2, the normalization ¯x = q/l , the reference frequency
ω0=pk/m, and the dimensionless timeτ = ω0t , but also the dimensionless damping
ratioζ = c/(2pkm), Eq. (2.5) can be normalized as
I∂ 2¯x ∂τ2+ 2ζAη £ η¤∂¯x ∂τ+ Ak¯x = 0. (2.7)
Eq. (2.7) shows that the damping is determined by a factor two and by two scaling pa-rameters (i.e.,η and ζ). To prevent this ambiguity, we introduce Aξ[ξ] = 2ζAη£
η¤. The
structure of Aξ[ξ] ∈ R6×6is equal to Aη£
η¤ (see Eq. (2.6)) except that the parameterη is replaced by the parameterξ. Subsequently, if we assume a solution in the form of ¯x = xeγτand substitute this into Eq. (2.7), the eigenproblem to determine the complex eigenvaluesγn∈ C and corresponding mode shapes xn∈ C6as a function ofξ, can be given by
¡
γ2
nI + γnAξ[ξ] + Ak¢ xn= 0 for n = 1, . . . , 6. (2.8) The structural eigenvalues corresponding to the mode shapes xn are, with the intro-duced dimensionless timeτ, denoted by λn= γnω0.
For allξ, the first eigensolution of this system is a rigid body solution (i.e., γ1= 0 ∀ ξ). Fig.2.5shows the evolution of the real and imaginary part of eigenvaluesγj for
2.3.LOCAL STRUCTURAL DAMPING CHANGES
2
21j = 2,...,6 as a function of the damping parameter ξ (i.e., ℜ(γj) [ξ] and ℑ(γj) [ξ], respec-tively). In the remainder of this section, we consider the more intuitive subdivision of the complex eigenvaluesγj. That is,
γj= ℜ ¡
γj¢ + iℑ¡γj¢ = σj+ i ωd , j, (2.9) where ℜ and ℑ indicate the real and imaginary part, respectively, i is the imaginary unit,
σjrepresents the modal decay, andωd , j represents the damped eigenvalue indicated by the subscript d . Although the modal decayσj is negative for dynamic stability reasons, we present the absolute of ℜ(γj) in Fig.2.5to be more intuitive.
|ℜ(γ2)| |ℜ(γ3)| |ℜ(γ4)| |ℜ(γ5)| |ℜ(γ6)| ℑ(γ2) ℑ(γ3) ℑ(γ4) ℑ(γ5) ℑ(γ6) ¯ ¯ℜ ¡ γj ¢¯ ¯ ξ ¿ 1 ξ ∼ 1 ξ À 1 10-4 10-2 100 ℑ¡ γj ¢ 10-2 100 102 ξ ¿ 1 ξ ∼ 1 ξ À 1 10-0.5 100 100.5 ξ
Figure 2.5: Evolution of the real (i.e., ℜ(γj); in top plot) and imaginary (i.e., ℑ(γj); in bottom plot) part of the 2ndto 6steigenvalueγjon a bi-logarithmic scale as a function ofξ (note: we show the absolute of ℜ(γj) to be more intuitive). These eigenvalues determine, with the reference angular velocityω0, the structural non-zero
