Micro-Electro-Mechanical-System (MEMS) technology represents one of the major breakthroughs of the last century. The miniaturization of devices like e.g., sensors, actuators, and their integration with electronic boards allow introducing them in everyday instruments, like e.g., smartphones, computers, cars, gaming consoles, as well as in cutting-edge technologies like e.g., augmented reality glasses, prostheses, self-driving vehicles etc. This technology and its applications are possible only if a MEMS device has predictable behaviour and a reliable working principle. For both requirements the MEMS designer performs numerical simulations to assess the performance before the fabrication. Nevertheless, the micro-scale poses challenges, rarely observed at the mesoscale, that cannot be ignored. The low damping factors due to the near-vacuum encapsulation of most MEMS devices and/or the large displacements and rotations needed for a good input-output signal ratio lead to a nonlinear behaviour of the device itself. Predicting the nonlinear dynamic response of a device represents a challenging task due to the long computational time needed with standard simulation methods like e.g. the finite element method, non-linear interactions between modes and the existence of alternative states that onset from bifurcations. Instead of tackling these problems with expensive full order methods, it is possible to resort to suitable reduced-order modeling techniques i.e. methods that allow reducing the dimensionality of the original problem. This thesis contributes to the state-of-art of MEMS nonlinear dynamics analysis by presenting analytical models and reduced-order models tailored for MEMS resonators, internal resonances, and bifurcations. After a short introduction to MEMS technology, nonlinearities, and modelling we present analytical techniques for internal resonances in MEMS. Starting from the normal forms of 1:2 and 1:3 internal resonances, multiple scales method solutions are developed and tailored for MEMS-like problems. Thanks to this analytical approach not only the system response in the conservative and non-conservative cases is studied but also the bifurcation portrait is analysed. A focus is put on the so-called Neimark-Sacker bifurcation that generates a quasi-periodic regime with an associated frequency comb in the power spectra of the response. Later, two reduced-order modelling strategies are considered: the implicit condensation and the proper orthogonal decomposition. Both allow deriving a reduced-order model in the form of a system of nonlinear ordinary differential equations that reproduces the MEMS dynamics. The former procedure, referring to geometric nonlinearities, uses a series of loads statically applied to the structure in order to identify the stress manifold of the system using one or very few low-frequency modes as a reduced basis. A similar approach can be used to reduce electrostatic forces contributions. First the theoretical settings and the algorithm are detailed and next several applications are considered addressing internal resonances, bifurcations and frequency comb. The results are validated against numerical and experimental data. Despite the flexibility offered by the implicit condensation method, the approach is limited in its application to moderate transformations. In systems where large rotations are present, e.g. micromirrors, this approach fails. As an alternative, we consider the proper orthogonal decomposition method. This technique is a linear reduced-order modelling method, which creates ad-hoc basis thanks to matrices of snapshots of the solutions. Even though the proper orthogonal decomposition is well-known in the literature, its application to the fields of MEMS is rather novel and reveals a remarkable performance. This contribution highlights both advantages and possible drawbacks, and proposes an in-depth analysis.
La tecnologia dei sistemi micro-elettro-meccanici o MEMS rappresenta una delle più grandi scoperte dell'ultimo secolo. La miniaturizzazione di dispositivi come sensori e attuatori e la loro integrazione in circuiti elettronici consente la loro introduzione negli strumenti della quotidianità (ad esempio smartphone, computer, console per videogiochi) oltre che in applicazioni d'avanguardia (ad esempio occhiali per la realtà aumentata, protesi, veicoli a guida automatica ecc.). Questa tecnologia e le sue applicazioni sono possibili solo se il dispositivo MEMS ha un comportamento prevedibile e un principio di funzionamento affidabile. Per entrambe queste necessità il designer MEMS effettua delle simulazioni numeriche per accertare le prestazioni prima della fabbricazione. Ciononostante la micro-scala pone delle sfide, osservabili raramente alla meso-scala, che non possono essere trascurate. I livelli di smorzamento bassi derivanti da un incapsulamento sottovuoto di molti dispositivi MEMS e/o i grandi spostamenti e rotazioni necessari per avere un buon rapporto tra segnale in ingresso o in uscita porta a comportamenti non lineari del dispositivo stesso. Prevedere la risposta dinamica non lineare del dispositivo rappresenta un compito arduo a causa dei tempi di calcolo richiesti da metodi di simulazione standard (es. il metodo degli elementi finiti), interazioni non lineari tra i modi e l'esistenza di soluzioni alterative derivanti dalle biforcazioni. Anziché affrontare questi problemi con costosi modelli ad alta fedeltà, è possibile impiegare una modellazione di ordine ridotto ossia metodi che permettono di ridurre la dimensione del problema originale. Questa tesi contribuisce allo stato dell'arte dell'analisi della dinamica non lineare nei MEMS proponendo modelli analitici e modelli di ordine ridotto specifici per risonatori MEMS, risonanze interne e biforcazioni. Dopo una breve introduzione alla tecnologia MEMS, le loro non linearità e la loro modellazione vengono presentati modelli analitici per risonanze interne nei MEMS. A partire dalle forme normali delle risonanze 1:2 e 1:3, si applica il metodo delle scale multiple specializzandolo a problemi nei MEMS. Grazie a questo approccio analitico viene derivata non solo la risposta del sistema in condizioni conservative e non ma viene analizzato anche il quadro delle biforcazioni del sistema. Viene studiata in particolare la cosiddetta biforcazione Neimark-Sacker che è alla base dei regimi quasi periodici che generano pettini di frequenza nello spettro delle frequenze della risposta. In seguito, sono considerate due strategie per modellazione di ordine ridotto: il metodo per condensazione implicita e la decomposizione ortogonale propria. Entrambe permettono di ricavare modelli ridotti dati da un sistema non lineare di equazioni differenziali ordinarie in grado di riprodurre la dinamica del dispositivo MEMS. La prima tecnica, considerando non linearità geometriche, utilizza una serie di carichi statici sulla struttura per identificare la varietà degli sforzi del sistema utilizzando uno o un numero limitato di modi a bassa frequenza come basi ridotte. Un approccio analogo può essere utilizzato per il contributo delle forze elettrostatiche. Viene mostrata l'impostazione teorica del metodo e l'algoritmo corrispondente, nel seguito sono mostrati vari esempi di applicazione. Con questo metodo sono modellate risonanze interne, biforcazioni e pettini di frequenza. I risultati sono validati sia numericamente che sperimentalmente. Nonostante la flessibilità del metodo per condensazione implicita, l'approccio risulta limitato a solo trasformazioni moderate. Il metodo è inapplicabile in sistemi dove sono presenti grandi rotazioni come nei microspecchi. In alternativa si considera il metodo per decomposizione ortogonale propria (POD). Questo è un metodo di riduzione d'ordine lineare che crea delle basi dedicate partendo da matrici di istantanee della soluzione. Nonostante la POD sia ben nota in letteratura, l'applicazione ai MEMS è relativamente nuova e dimostra prestazioni notevoli. Questo contributo evidenzia sia i vantaggi che i possibili svantaggi proponendo un'analisi approfondita.
Reduced order modelling and analysis of internal resonances, bifurcations, and frequency combs in MEMS
Gobat, Giorgio
2021/2022
Abstract
Micro-Electro-Mechanical-System (MEMS) technology represents one of the major breakthroughs of the last century. The miniaturization of devices like e.g., sensors, actuators, and their integration with electronic boards allow introducing them in everyday instruments, like e.g., smartphones, computers, cars, gaming consoles, as well as in cutting-edge technologies like e.g., augmented reality glasses, prostheses, self-driving vehicles etc. This technology and its applications are possible only if a MEMS device has predictable behaviour and a reliable working principle. For both requirements the MEMS designer performs numerical simulations to assess the performance before the fabrication. Nevertheless, the micro-scale poses challenges, rarely observed at the mesoscale, that cannot be ignored. The low damping factors due to the near-vacuum encapsulation of most MEMS devices and/or the large displacements and rotations needed for a good input-output signal ratio lead to a nonlinear behaviour of the device itself. Predicting the nonlinear dynamic response of a device represents a challenging task due to the long computational time needed with standard simulation methods like e.g. the finite element method, non-linear interactions between modes and the existence of alternative states that onset from bifurcations. Instead of tackling these problems with expensive full order methods, it is possible to resort to suitable reduced-order modeling techniques i.e. methods that allow reducing the dimensionality of the original problem. This thesis contributes to the state-of-art of MEMS nonlinear dynamics analysis by presenting analytical models and reduced-order models tailored for MEMS resonators, internal resonances, and bifurcations. After a short introduction to MEMS technology, nonlinearities, and modelling we present analytical techniques for internal resonances in MEMS. Starting from the normal forms of 1:2 and 1:3 internal resonances, multiple scales method solutions are developed and tailored for MEMS-like problems. Thanks to this analytical approach not only the system response in the conservative and non-conservative cases is studied but also the bifurcation portrait is analysed. A focus is put on the so-called Neimark-Sacker bifurcation that generates a quasi-periodic regime with an associated frequency comb in the power spectra of the response. Later, two reduced-order modelling strategies are considered: the implicit condensation and the proper orthogonal decomposition. Both allow deriving a reduced-order model in the form of a system of nonlinear ordinary differential equations that reproduces the MEMS dynamics. The former procedure, referring to geometric nonlinearities, uses a series of loads statically applied to the structure in order to identify the stress manifold of the system using one or very few low-frequency modes as a reduced basis. A similar approach can be used to reduce electrostatic forces contributions. First the theoretical settings and the algorithm are detailed and next several applications are considered addressing internal resonances, bifurcations and frequency comb. The results are validated against numerical and experimental data. Despite the flexibility offered by the implicit condensation method, the approach is limited in its application to moderate transformations. In systems where large rotations are present, e.g. micromirrors, this approach fails. As an alternative, we consider the proper orthogonal decomposition method. This technique is a linear reduced-order modelling method, which creates ad-hoc basis thanks to matrices of snapshots of the solutions. Even though the proper orthogonal decomposition is well-known in the literature, its application to the fields of MEMS is rather novel and reveals a remarkable performance. This contribution highlights both advantages and possible drawbacks, and proposes an in-depth analysis.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/189712