This paper delves into the static and dynamic behavior of graphene cantilever beam resonators under electrostatic actuation at their free tips. A rigorous analysis of the system’s response is performed. The constitutive nonlinear equation of the system is derived using the energy method and Hamilton’s principle. An analytical solution to the nonlinear static problem is obtained. The generalized stiffness coefficient for the lumped model of the cantilever graphene beam under load at its tip is calculated, enabling a comprehensive analysis of its dynamic behavior. A key focus is on investigating the dynamic pull-in conditions of the system under both constant and harmonic excitation. Analytical predictions are validated through numerical simulations. The system exhibits periodic solutions when the excitation parameters are below a certain threshold described by a separatrix curve, leading to sustained oscillations. On the other hand, if the excitation parameters exceed this threshold, the system experiences pull-in instability, causing the beam to touch down. Furthermore, we explore the impact of excitation frequency on the dynamic response of the graphene cantilever beam under harmonic load. The simulations reveal that choosing the excitation frequency near the beam’s resonance frequency can lead to structural collapse under certain parameter conditions.

The initial value problem for a lumped parameter model arising from design of magneto–electromechanical device with a current-carrying conductor is analyzed. The differential equation is nonlinear because it includes the magnetic force term. The analysis for the dynamic pull-in occurring in the system is presented. The pull-in threshold is given analytically in terms of model parameters. Sufficient conditions for the existence of periodic solutions are proved analytically and verified numerically. The results can be useful for understanding and design of one-degree-of-freedom models of magnetically actuated beams.

Research of microstructure and permeability evolution of coal following LN₂ treatment elucidate the process of cryogenic fracturing due to environmentally friendly behavior in comparison with conventional hydraulic fracturing. The evolution of the 2D microstructure of bituminous coal before and after LN₂ treatment was examined using a high-resolution camera. The image processing was implemented using functions from the OpenCV Python library that are sequentially applied to digital images of original coal samples. The images were converted into binary pixel matrices to identify cracks and to evaluate the number of cracks, crack density, total crack area, and average crack length. Results were visualized using Seaborn and Matplotlib Python libraries. There were calculations of total crack area (TCA), total number of cracks (TNC), crack density (CD), the average length of cracks (Q2), first (Q1) and third (Q3) quartiles in fracture length statistics. Our findings demonstrate a progressive increase in the Total Crack Area (δTCA) with longer freezing times and an increased number of freezing–thawing cycles. In contrast, the change in crack density (δCD) was generally unaffected by freezing time alone but exhibited a significant increase after several freezing–thawing cycles. Among the freezing times investigated, the highest crack density (CD) value of 300 m⁻¹ was achieved in FT60, while the lowest CD value of 31.25 m⁻¹ was observed in FT90 after liquid nitrogen (LN2) treatment. Additionally, the FTC4 process resulted in a 50% augmentation in the number of cracks, whereas the FTC5 process tripled the number of small cracks.