Non-stationary resonance dynamics of weakly coupled pendula

In this paper we fill the gap in understanding the non-stationary resonance dynamics of the weakly coupled pendula model, having significant applications in numerous fields of physics such as super- conducting Josephson junctions, Bose-Einstein condensates, DNA, etc.. While common knowledge of the problem is based on two alternative limiting asymptotics, namely the quasi-linear approach and the approximation of independent pendula, we present a unified description in the framework of new concept of Limiting Phase Trajectories (LPT), without any restriction on the amplitudes of oscillation. As a result the conditions of intense energy exchange between the pendula and transition to energy localization are revealed in all possible diapason of initial conditions. By doing so, the roots and the domain of chaotic behavior are clarified as they are associated with this transition while simultaneously approaching the pendulum separatrix. The analytical findings are corrobo- rated by numerical simulations. By considering the simplest case of two weakly coupled pendula, we pave the ground for new opening possibilities of significant extensions in both fundamental and applied directions.Comment: 7 pages, 7 figure

Semi-Inverse Method in the Nonlinear Dynamics

The semi-inverse method based on using an internal small parameter of the nonlinear problems is discussed on the examples of the chain of coupled pendula and of the forced pendulum. The efficiency of such approach is highly appeared when the non-stationary dynamical problems are considered. In the framework of this method we demonstrate that both the spectrum of nonlinear normal modes and the interaction of them can be analysed successfully

Vibrations of Carbon Nanotubes: nonlinear models and energy distribution

Vibrations of Single-Walled Carbon Nanotubes for various boundary conditions are considered in the framework of the Sanders-Koiter thin shell theory. A double series expansion of displacement fields, based on the Chebyshev orthogonal polynomials and harmonic functions, is used to analyse numerically the natural frequencies of shells having free or clamped edges. A reduced form of the Sanders-Koiter theory is developed by assuming small circumferential and shear deformations; such approach allows to determine an analytical solution for the natural frequencies. The numerical model is validated with the results of molecular dynamics and finite element analyses present in literature. The analytical model is validated by means of comparisons with the numerical approach. Nonlinear vibrations and energy distribution of carbon nanotubes are then considered

Nonlinear vibrations and energy distribution of Single-Walled Carbon Nanotubes

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The Rayleigh-Ritz method is applied in order to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations. The energy distribution of the system is studied by considering combinations of different vibration modes. The effect of the conjugate modes participation on the energy distribution is analysed

Nonlinear Dynamics of Single-Walled Carbon Nanotubes

The nonlinear dynamics of Single-Walled Carbon Nanotubes is studied. The Sanders-Koiter elastic shell theory is applied. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. Free boundary conditions are considered. The total energy distribution of the system is studied by considering the combinations of different vibration modes. The effect of the companion mode participation on the energy distribution is analysed

Nonlinear dynamics of Single-Walled Carbon Nanotubes

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The Rayleigh-Ritz method is applied to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations. The total energy distribution of the shell is studied by considering combinations of different vibration modes. The effect of the conjugate modes is analysed

Nonlinear vibrations and energy distribution of carbon nanotubes

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter thin shell theory is applied in order to obtain the elastic strain and kinetic energy. The carbon nanotube deformation is described in terms of axial, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The Rayleigh-Ritz method is applied to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is then considered to obtain a set of nonlinear ordinary differential equations. The total energy distribution of the shell is studied by considering combinations of different vibration modes. The effect of the conjugate modes is analysed

Nonlinear vibrations and energy exchange of single-walled carbon nanotubes. Radial breathing modes

In this paper, the nonlinear vibrations and energy exchange of single-walled carbon nanotubes (SWNTs) are analysed. The Sanders-Koiter shell theory is used to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary conditions are applied. The resonant interaction between radial breathing (axisymmetric) modes (RBMs) is analysed. An energy method, based on the Lagrange equations, is considered in order to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is then solved applying the implicit Runge-Kutta numerical method. The present model is validated in linear field comparing the RBM natural frequencies numerically predicted with data reported in the literature from experiments and molecular dynamics simulations. The nonlinear energy exchange between the two halves along the SWNT axis in the time is studied for different amplitudes of initial excitation applied to the two lowest frequency resonant RBMs. The influence of the SWNT aspect ratio on the numerical value of the nonlinear energy beating period under different boundary conditions is analysed