Nonlinear dynamic simulations of mechanical resonators have been facilitated by the advent of computational techniques that generate nonlinear reduced order models (ROMs) using the finite element (FE) method. However, designing devices with specific nonlinear characteristics remains inefficient since it requires manual adjustment of the design parameters and can result in suboptimal designs. Here, we integrate an FE-based nonlinear ROM technique with a derivative-free optimization algorithm to enable the design of nonlinear mechanical resonators. The resulting methodology is used to optimize the support design of high-stress nanomechanical Si ₃N ₄ string resonators, in the presence of conflicting objectives such as simultaneous enhancement of Q-factor and nonlinear Duffing constant. To that end, we generate Pareto frontiers that highlight the trade-offs between optimization objectives and validate the results both numerically and experimentally. To further demonstrate the capability of multi-objective optimization for practical design challenges, we simultaneously optimize the design of nanoresonators for three key figure-of-merits in resonant sensing: power consumption, sensitivity and response time. The presented methodology can facilitate and accelerate designing (nano) mechanical resonators with optimized performance for a wide variety of applications. (Figure presented.)

Most micro- and nanomechanical devices are designed to operate within the linear dynamic range by using simple geometries, primarily due to the limited knowledge in utilizing nonlinearity and constraints in fabrication techniques. However, it is anticipated that the scope of applications and fundamental research can be significantly expanded, if these tiny systems can be precisely engineered to account for and exploit their various nonlinear dynamic behaviors. This thesis provides a comprehensive study on the optimization of dynamical properties of high-Q nanomechanical resonators, spanning from linear to nonlinear dynamics and evolving from single-mode to multi-mode analysis.

We first give an introduction to the development of micro- and nanomechanical resonators in Chapter 1. We focus on their unique mechanics, including very low dissipation and strong nonlinearities. Furthermore, we elaborate on the motivation behind this thesis and the need for linking engineering optimization with micro- and nanomechanical resonators. Followed by Chapter 2, we elaborate on the methodology we use throughout this thesis, including fabrication techniques of nanomechanical Si3N4 devices, characterization approaches by optical measurements, and modeling procedures for structural dynamics. Among all methodologies, we highlight the Finite Element (FE)- based Reduced Order Models (ROMs) that can accurately capture the geometric details and boundary conditions, which facilitate the design of resonators with predictable dynamical properties.

We start from investigating linear dynamics of Si3N4 nanostrings in Chapter 3, where the tuning effects of their soft-clamping supports on resonance frequency and Q-factor are evaluated. We experimentally and theoretically reveal a trade-off between maintaining high stress and low stiffness of the supports in designing high-Q resonators fabricated with initial strain. By optimizing this trade-off with our soft-clamping design, we obtain a 50% enhancement of Q-factor compared to doubly-clamped string resonators. With stronger drive levels, in Chapter 4, we show that the nonlinear dynamics can also be substantially tailored by soft-clamping supports. Through careful engineering of support geometries, we introduce softening nonlinearity by stress-induced buckling, allowing precise control over the nonlinear dynamic responses in doubly supported nanostrings, which conventionally exhibit hardening behaviors.

Based on the accurate modeling of both linear and nonlinear dynamics that is validated by experiments, we integrate our FE-based ROM technique with a derivative-free optimization algorithm for the design of nonlinear mechanical resonators in Chapter 5. By optimizing the support’s geometry of our nanostrings, we show that the proposed methodology is not only capable of handling a single optimization goal, but also multiple conflicting objectives, such as the simultaneous enhancement of Q-factor and the Duffing constant. Besides, we generate Pareto frontiers that visualize the trade-offs among multiple optimization objectives, verify the optimized results with brute-force simulations and validate the numerical framework with experiments.

Apart from the dynamics in a single mode, we observe modal interactions between multiple vibrational modes of our nanostrings in the strong nonlinear regime. In Chapter 6, we demonstrate that soft-clamping techniques, commonly utilized to achieve high-Q resonators, can be employed to engineer mode coupling. We verify the analytically derived two-degree-of-freedom system between the lowest two out-of-plane modes by FE-based ROMs and experiments. We further reveal the significant impact of multi-mode interactions on the nanostrings’ frequency response, demonstrating additional opportunities to tailor the nonlinear dynamics of mechanical resonators facilitated by soft clamping. Moreover, we highlight the design potential of soft-clamping supports through the geometric optimization of two-mode coupling, showcasing the effective Duffing constant of the driven mode can be increased by 70%, as well as the onset of mode coupling can be geometrically programmed to either facilitate or inhibit its occurrence.

We conclude all works presented for achieving the optimization of nonlinear dynamics in nanomechanical resonators, and give an outlook for future directions in Chapter 7.

Although strain engineering and soft-clamping techniques for attaining high Q-factors in nanoresonators have received much attention, their impact on nonlinear dynamics is not fully understood. In this study, we show that nonlinearity of high-Q Si₃N₄ nanomechanical string resonators can be substantially tuned by support design. Through careful engineering of support geometries, we control both stress and mechanical nonlinearities, effectively tuning nonlinear stiffness of two orders of magnitude. Our approach also allows control over the sign of the Duffing constant resulting in nonlinear softening of the mechanical mode that conventionally exhibits hardening behavior. We elucidate the influence of support design on the magnitude and trend of the nonlinearity using both analytical and finite element-based reduced-order models that validate our experimental findings. Our work provides evidence of the role of soft-clamping on the nonlinear dynamic response of nanoresonators, offering an alternative pathway for nullifying or enhancing nonlinearity in a reproducible and passive manner.

In recent years, the Q-factor of Si 3 N 4 nanomechanical resonators has significantly been increased by soft-clamping techniques using large and complex support structures. To date, however, obtaining similar performance with smaller supports has remained a challenge. Here, we make use of torsion beam supports to tune the Q-factor of Si 3 N 4 string resonators. By design optimization of the supports, we obtain a 50% Q-factor enhancement compared to the standard clamped-clamped string resonators. By performing experimental and numerical studies, we show that further improvement of the Q-factor is limited by a trade-off between maximizing stress and minimizing torsional support stiffness. Thus, our study also provides insight into dissipation limits of high-stress string resonators and outlines how advanced designs can be realized for reaching ultimate f 0 × Q product while maintaining a small footprint.