# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 1 Introduction  
The interaction of rarefied gas flow with micro-cantilevers is an area of growing interest in the fields of microtechnology and fluid dynamics. The study of interactions between fluid and mechanical fields is crucial for advancing applications such as sensors, actuators, and energy-harvesting devices where the performance can be significantly influenced by the surrounding gas flow characteristics [1, 2]. Theoretical models have been developed to analyse the dynamic behaviour of micro-cantilevers taking into account factors such as particle attachment and the influence of external forces. In [2], the rectangular micro-cantilever is regarded as a point mass in the dynamic model of forced and damped harmonic oscillators. The authors use single- and bimodal-frequency excitation schemes for actuation of the micro-cantilever in the presence of dynamic acoustic forces. Traditionally, fluid-structure interactions have been examined using continuum models like the Navier-Stokes equations. However, these can lose accuracy when the character size of the object is comparable to the mean free path of molecules, leading to important non-equilibrium effects, especially when the boundary layer plays an important role. Forinstance, it was proved by [PERSON] et al. [3] that in cavity flow simulations, the difference between Navier-Stokes and Direct Simulation Monte Carlo (DSMC) solutions becomes more notable near the cavity's corner tips, where local non-equilibrium effects in the boundary layer become dominant even at \(\text{Kn}=0.005\), a value nearing the continuum regime. Their research has shown that the validity of Navier-Stokes equations can depend significantly on the specific problem, challenging general assumptions about the range of their applicability. To address this, the authors of this study have applied the DSMC to model the rarefied gas flow, which is especially reliable in the slip-flow and transition regimes [4; 5; 6]. DSMC, initially developed by [PERSON] in the 1960s [7], is a molecular-based benchmark method that takes place between molecular dynamics--whose computational requirements for these sizes are often impossible and continuum models, which miss important non-equilibrium effects. The understanding of how rarefied gas flow interacts with micro-cantilevers is further enhanced by experimental investigations that reveal the effects of varying flow rates and gas properties on the vibration characteristics of these structures. Such studies are essential for optimizing the design and functionality of micro-cantilevers in real-world applications, particularly in environments where gas flow is not well approximated by classical fluid dynamics [8; 9]. Moreover, the phenomenon of vortex shedding can also play a significant role in the dynamics of micro-cantilevers. Vortex-induced vibration (VIV) is a phenomenon that arises when fluid flow interacts with structures, leading to oscillations that can have significant implications for engineering applications. This phenomenon is characterized by the periodic detachment of vortices from the surface of a body, which generates fluctuating forces that can induce vibrations in structures such as bridges, pipelines, and offshore platforms. Understanding VIV is essential for the design and maintenance of these structures, as it can lead to fatigue failure and catastrophic failures if not properly managed [10; 11; 12]. The mechanisms behind VIV are complex, involving fluid-structure interactions (FSI) that can result in both self-excited and forced vibrations. When the frequency of vortex shedding aligns closely with the natural frequency of a structure, a condition known as \"lock-in\" occurs, amplifying the vibrations and potentially leading to severe structural damage [13; 14]. This phenomenon is particularly relevant in the context of marine engineering, where structures such as risers and cables are subjected to dynamic loading from ocean currents [15; 16; 11].