Lev Shtukin

In this work, we study the nonlinear dynamics of a mode-localized mass detector. A system of equations is obtained for two weakly coupled beam resonators with an alternating electric current flowing through them and taking into account the point mass on one of the resonators. The onedimensional problem of thermal conductivity is solved, and a steady-state harmonic temperature distribution in the volume of the resonators is obtained. Using the method of multiple scales, a system of equations in slow variables is obtained, on the basis of which instability zones of parametric resonance, amplitude-frequency characteristics, as well as zones of attraction of various branches, are found. It is shown that in a completely symmetrical system (without a deposited particle), the effect of branching of the antiphase branch of the frequency response is observed, which leads to the existence of an oscillation regime with different amplitudes in a certain frequency range. In the presence of a deposited particle, this effect is enhanced, and the branching point and the ratio of the amplitudes of oscillations of the resonators depend on the mass of the deposited particle.

In this work, we construct and analyze a mathematical model of coupled longitudinaltransverse vibrations of a rectangular pretensioned strip under conditions of internal combinational resonance between two transverse modes and one longitudinal vibration mode of the structure in view of advanced nanosystems made of two-dimensional materials. The issues of generating frequency combs based on the proposed nanoresonator design are investigated in the context of the development of nondestructive methods of laser optothermal excitation of vibrations of atomically thin nanostructures. Conditions have been analytically found for the amount of deformation of the initial tension of the layer required to realize resonance between eigenmodes with given indices of variability along the length. The necessary relationships between the indices of vibration modes involved in nonlinear interaction are determined. It is shown that, under conditions of internal resonance, beats are excited in the system, the spectrum of which has the form of a frequency comb. Two qualitatively different types of beats are identified — those caused by the initial excitation in the working longitudinal form of vibrations and in two transverse forms. A significant dependence of the spectral composition of the generated frequency combs on the relationships between the amplitudes of the initial disturbance along three interacting modal coordinates and on the value of the internal frequency detuning parameter of the system is shown.