©2020 Chaitanya Borra All Rights Reserved

©2020 CHAITANYA BORRA ALL RIGHTS RESERVED DYNAMICS OF LARGE ARRAY MICRO/NANO RESONATORS A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Chaitanya Borra May, 2020 DYNAMICS OF LARGE ARRAY MICRO/NANO RESONATORS Chaitanya Borra Dissertation Approved: Accepted: Advisor Department Chair D. Dane Quinn Sergio Felicelli Committee Member Dean of the College Graham Kelly Craig Menzemer Committee Member Dean of the Graduate School Ernian Pan Marnie Saunders Committee Member Date Patrick Wilber Committee Member Kwek Tze Tan ii ABSTRACT This work describes an analytical framework suitable for the analysis of large-scale arrays of coupled resonators, including those which feature amplitude and phase dynamics, inherent element-level parameter variation, nonlinearity, and/or noise. In particular, this analysis allows for the consideration of coupled systems in which the number of individual resonators is large, extending as far as the continuum limit corresponding to an infinite number of resonators. Moreover, this framework per- mits analytical predictions for the amplitude and phase dynamics of such systems. The utility of this analytical methodology is explored through the analysis of a system of N non-identical resonators with global coupling, including both reactive and dissipative components, physically motivated by an electromagnetically-transduced microresonator array. In addition to the amplitude and phase dynamics, the behavior of the system as the number of resonators varies is investigated and the convergence of the discrete system to the infinite-N limit is characterized. The equilibrium obtained by the array of resonators may be unique if the behavior of many identical or slightly different resonators in the population is small, such equilibrium is also called mono-stable equilibrium. The dynamics of such systems can be identified with a typical single degree of freedom systems under the iii influence of harmonic excitations at different frequencies. The behavior ofanarray of resonators can be described using two self-consistency variables and a force, which not only includes the external excitation but also due to the reactive and dissipative coupling parameters, this is called a global coupling function. The behavior of self-consistency variables and global coupling forces in the case of a linear system is largely monotonous for the variations in the coupling parameters. Unlike monostable and linear systems, the effect of reactive and dissipative coupling on non-linear systems cannot be decoupled and is not so straightforward. Duffing like behavior can be observed by increasing the external excitations ornon- linear stiffness. In order to study the effect of reactive coupling, on these systems, the dissipative coupling is held constant and reactive coupling is varied, it is observed that multiple solutions start appearing when the reactive coupling is increased. This is mainly due to the resonators with their detuning close to the resonant actuating frequency, this localized phenomenon is similar to a Duffing response. Two different bifurcations resulting from the variation of coupling parameters have been identified, which explains the underlying phenomenon that leads to multiple-equilibrium distributions. The stability of distributions is found by evaluating the eigenvalues, from the discrete spectrum equation. iv DEDICATION I would like to dedicate this work to my Grandfather(Surya Prakasa Rao), my Par- ents(Rama Mohan and Manikya Kumari), my wife(Lakshmi Kalyani) and my younger brother(Vamsi). v ACKNOWLEDGEMENTS I would like to take this opportunity to thank my advisor Dr. Dane. D. Quinn for his guidance during my research. I am grateful to Dr. Graham Kelly, Dr. Kwek Tze Tan, Dr. Ernian Pan, and Dr. Patrick Wilber for serving on my committee. I am very thankful to National Science Foundation for supporting my dissertation research (Grant number 1537701). I am also thankful to Dr. Quinn and Dr.Sergio Felicelli for providing the funding and assistantship opportunity. A special thanks to my fellow students Allen, Ali, Dushyanth, Sunny and others whose friendships have made my experience at the University of Akron more enjoyable. I would also like to express my appreciation to the current and former staff of the Mechanical Engineering and Aerospace department including Shannon, Ellen, Stacy and Lisa, for their help and friendship during my tenure at the University of Akron. I would like to thank my family and friends for their assistance during my academic career. Finally, and most importantly, I would like to express my deepest gratitude to my parents, wife and my brother whose love, support and encouragement made this dissertation possible. vi TABLE OF CONTENTS Page LIST OF FIGURES . ix CHAPTER I. INTRODUCTION . 1 1.1 Single Resonator Dynamics . 4 1.2 Coupled resonators . 11 1.3 Organization of the Thesis . 19 II. LITERATURE . 22 2.1 Coupled Resonators . 24 2.2 Mathematical models . 29 III. ANALYTICAL MODELING OF LARGE ARRAY RESONATORS . 39 3.1 A Discrete System of Coupled Resonators . 39 3.2 Continuum Model . 41 3.3 Results . 49 IV. MONOSTABLE DISTRIBUTIONS . 56 4.1 Introduction . 56 4.2 Results . 57 4.3 Conclusions . 67 vii V. EFFECT OF COUPLING PARAMETERS ON MULTIPLE EQUILIBRIUM STATES . 68 5.1 Monostable-Unique Solutions . 69 5.2 Multiple Equilibrium Distributions . 71 5.3 Non-isolated Equilibrium states . 101 5.4 Stability region . 125 5.5 Summary . 130 VI. CONCLUSION AND FUTURE WORK . 134 6.1 Summary . 134 6.2 Suggested Future work . 139 BIBLIOGRAPHY . 140 viii LIST OF FIGURES Figure Page 1.1 Schematic diagram of beam element used as resonator, here u, v, and are longitudinal, transverse and angular displacements of the differential beam element [1]. .5 1.2 F0 = 2:0 ; ζ = 0:1 ; γ = 0:1 a)Amplitude R(σ) vs σ b) Phase ϕ(σ) vs σ 9 1.3 F0 = 2:0 ; ζ = 0:1 a)Amplitude R(σ) vs σ b) Phase ϕ(σ) vs σ, γ = −0:5, γ = −0:1, γ = 0, γ = 0:1, γ = 0:5. 10 1.4 γ = 0:1 ; ζ = 0:1 Frequency response R(σ) vs σ at F0 = 0:5, F0 = 1:0, F0 = 3:0, backbone curve representing the 3γR2 locus of peak amplitudes(σ = 8! ).................... 11 1.5 γ = 0:1 ;F0 = 2:0 Frequency response R(σ) vs σ at ζ = 0:05, ζ = 0:1, ζ = 0:2, ζ = 0:5..................... 12 1.6 Coupled system of resonators with 2 degree of freedom systems . 12 1.7 F0 = 2:0 ; ζ1 = ζ2 = 0:25 ; αg = 0 R(1)( σ1 = −5:0), R(2)( σ2 = 5:0) vs Ω (excitation frequency) for (a) βg = 0 , (b) βg = 0:5 , (c) βg = 5:0 and (d) βg = 10:0 . 16 1.8 F0 = 2:0 ; ζ1 = ζ2 = 0:25 ; βg = 0:5 R(1)( σ1 = −5:0), R(2)( σ2 = 5:0) vs Ω (excitation frequency) for (a) αg = 0 , (b) αg = 0:5 , (c) αg = 2:0 and (d) αg = 10:0 . 17 1.9 F0 = 2:0 ; ζ1 = ζ2 = 0:25 ; βg = 2:0 ; αg = 0:2 R(1)( b b σ1 = −5:0), R(2)( b b σ2 = 5:0) vs Ω (excitation frequency) for (a) γ = 0 , (b) γ = 0:08 , (c) γ = 0:5 and (d) γ = 2:0 . 18 2.1 SEM image of two resonant filter [2], series-coupled resonator deigned for operation at atmospheric pressure. 25 2.2 SEM image of polysilicon fabricated micromechanical disk-arrays [3],a) Top-view b) Iso-view image. 26 ix 2.3 SEM image of four 500µm × 4µm clamped-clamped array beams with 9µm spacing [4], through-support coupling was extended to large arrays simply placing beams adjacent to each other. 27 2.4 Scanning electron micrograph of a representative electromagnetically transduced microresonator. The device consists of an elec- trically and mechanically isolated silicon microcantilever and two Au/Cr wire loops, which follow the perimeter of the microcantilever [5]. 30 3.1 Response for a mono-stable solution γ = 0:02, bc bc , for coupling parameters (α ; β) = (0:5 ; 1:0). ...................... 50 3.2 R(s) vs m(s) for global coupling constants (V;W ) = (0:3840 ; −0:3298), given parameters γ = 0:2 ; (α ; β) = (0:5 ; 1:0) X(s) > 0, X(s) > 0(unstable branch), X(s) = 0, X(s) < 0 ................................ 51 bc bc E bc bc E U bc bc E 3.3 Multiple solutions at γ = 0:08, 6, 6 , 7, for coupling parameters (α ; β) = (0:5 ; 1:0) . 53 3.4 Re(λ) vs Im(λ), γ = 0:02 (α ; β) = (0:5 ; 1:0) bc bc Discrete spectrum. 54 E E U E 3.5 Spectrum plot Re(λ) vs Im(λ), γ = 0:08 a) 6, b) 6 and c) 7 , E E E U both 6 and 7 have eigenvalues with Re(λ) < 0; 6 has both the spectrums with Re(λ) > 0 ......................... 55 4.1 Linear response functions (α = 2:00, β = 5:00, γ = 0, N = 100); iterative solution, ⊗ ⊗ ⊗ direct numerical simulation; (a) R(s), (b) Φ(s).................................... 59 4.2 Coupling kernels (α = 2:00, β = 5:00, γ = 0, N = 100, iterative solution); (a) A(s) ρ(s), (b) B(s) ρ(s).................... 59 4.3 Linear response functions varying α (β = 10:00, γ = 0, N = 100); α = 0 ; α = 4:0 ; α = 8:0; (a)R(s), (b)Φ(s) . 60 4.4 Linear response functions varying β (α = 4:00, γ = 0, N = 100); β = 0 ; ; β = 10:0 ; β = 20:0; (a)R(s), (b)Φ(s) . 61 4.5 Maximum response amplitude Rmax, varying coupling parameters (γ = 0, N = 100); iterative solution, bC bC bC direct numerical simulation; (a) dissipative coupling (β = 0), (b) reactive coupling (α = 0).

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