Large-amplitude Nonlinear Vibrations Research Articles

Large-amplitude nonlinear vibrations characteristics of bi-directional functionally graded plate having microstructure defects in hygro-thermal environment

The present article investigates the nonlinear hygro-thermal vibration characteristics of bi-directional functionally graded plates (BFGP) with microstructural defects. The distribution of material properties is anticipated to vary gradually along both thickness and length directions according to the modified power-law distribution. A novel expression has been developed to solve for the nonlinear temperature variation along the thickness of the BFGP. The micro-structural defects in terms of porosity have been incorporated in the formulation using a newly developed porosity model for a BFGP. A refined non-polynomial trigonometric higher-order shear deformation theory (HSDT) with a von Karman sense of geometric nonlinearity has been employed for the vibration response of BFGP. The finite element formulation has been carried out utilizing the C0 continuous Lagrangian quadrilateral element with nine nodes and eight degrees of freedom per node. The equations of motion have been derived using a variational approach. Various validation and comparative studies have been conducted to substantiate the current formulation's authenticity. The effect of the geometric nonlinearity, boundary constraints, volume fraction index along thickness (n) and volume fraction index along length (q), porosity index, temperature, and moisture concentration difference on the vibration frequency ratios (VFR) has been covered in depth. The VFR is more sensitive to the volume fraction indices (VFI) ' n′ than 'q'. Incorporating porosity decreases the VFR of BFGP, but it increases at higher amplitude ratios when BFGP becomes more porous. The VFR of BFGP exhibits nonlinear behavior with increasing moisture concentration and also varies non-linearly with the rise in amplitude ratio due to the hygro-thermal environment.

Coexistence of bistable multi-pulse chaotic motions with large amplitude vibrations in buckled sandwich plate under transverse and in-plane excitations.

This paper researches the dynamic snap-through phenomena and the coexistence of the multi-pulse jumping chaotic dynamics in bistable equilibrium positions and the large amplitude nonlinear vibrations for a simply supported buckled 3D-kagome truss core sandwich rectangular plate under combined transverse and in-plane excitations based on the extended high-dimensional Melnikov method. According to the two-degree-of-freedom non-autonomous nonlinear dynamical system of the buckled truss core sandwich rectangular plate and the theory of normal form, it is found that some nonlinear terms have less effects on the nonlinear dynamic responses than other nonlinear terms. The extended high-dimensional Melnikov method is employed to investigate the dynamic snap-through phenomena and the bistable multi-pulse jumping chaotic vibrations around the upper-mode and the lower-mode for two different cases of buckling in the truss core sandwich rectangular plate. A numerical method is used to detect the bistable multi-pulse jumping chaotic vibrations around two stable equilibrium positions for the non-autonomous buckled truss core sandwich rectangular plate. The obtained theoretical and numerical results indicate that there exists the coexistence of the bistable multi-pulse jumping chaotic vibrations and the large amplitude nonlinear vibrations in the buckled truss core sandwich rectangular plate under combined transverse and in-plane excitations.

Nonlinear performance analysis of forced carbon nanotube-based bio-mass sensors

In this effort, an analytical solution is proposed for the large amplitude nonlinear vibrations of doubly clamped carbon nanotube (CNT)-based nano-scale bio-mass sensors. The single walled CNT is modeled as an elastic Euler–Bernoulli nano-scale beam and the size effects are introduced into the mathematical model of the system through Eringen’s nonlocal elastic field theory. The nonlinearity arises due to mid-plane stretching of the bridged CNT, and is accounted for as the von Karman nonlinearity. The impacts of deposited nano-scale bio-object, its geometrical properties, and its landing position along the longitudinal axis of the CNT-based resonator are considered. The nonlinear equations of motion are derived based on Hamilton’s principle and then the Method of Multiple Scales is employed to derive an analytical approximate solution for the system’s response. To verify the analytical solution and show its limits of applicability, the equations of motion are discretized by multi-mode Galerkin’s method and then the obtained set of equations are numerically solved by Runge–Kutta method and compared with those obtained by analytical solution. The potential applications of the CNT-based resonators for both of nonlinear frequency-/amplitude-based mass sensing are investigated and discussed. The obtained results show that the amplitude-based mass sensing has higher performance than frequency-based one in high quality-factor environments, such as in vitro biological mass sensing in the air or vacuum and inversely the frequency-based mass sensing method has higher mass sensibility in low quality factor environments, such as in vivo biological mass sensing in the liquid solution samples.

Soliton microdynamics of the generation of new-type nonlinear surface vibrations, dissociation, and surfing diffusion in diatomic crystals of the uranium nitride type

The microdynamics of large-amplitude nonlinear vibrations of uranium nitride diatomic lattices has been investigated using the computer simulation and neutron scattering methods at temperatures T = 600–2500°C near the thresholds of the dissociation and destruction of the reactor fuel materials. It has been found using the computer simulation that, in the spectral gap between the frequency bands of acoustic and optical phonons in crystals with an open surface, there are resonances of new-type harmonic surface vibrations and a gap-filling band of their genetic successors, i.e., nonlinear surface vibrations. Experimental measurements of the slow neutron scattering spectra of uranium nitride on the DIN-2PI neutron spectrometer have revealed resonances and bands of these surface vibrations in the spectral gap, as well as higher optical vibration overtones. It has been shown that the solitons and bisolitons initiate the formation and collapse of dynamic pores with the generation of surface vibrations at the boundaries of the cavities, evaporation of atoms and atomic clusters, formation of cracks, and destruction of the material. It has been demonstrated that the mass transfer of nitrogen in cracks and along grain boundaries can occur through the revealed microdynamics mechanism of the surfing diffusion of light nitrogen atoms at large-amplitude soliton waves propagating in the stabilizing sublattice of heavy uranium atoms and in the nitrogen sublattice.

Computational models for large amplitude nonlinear vibrations of electrostatically actuated carbon nanotube-based mass sensors

A comprehensive multiphysics model of a cantilevered carbon nanotube (CNT) including geometric and electrostatic nonlinearities is developed. The continuous model is reduced to a finite degree of freedom system by the Galerkin discretization and solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). The influence of higher modes on the nonlinear dynamics of the considered resonator is investigated in order to retain the number of modes which will be used by the HBM+ANM procedure. Several simulations are performed for a specific CNT design in order to obtain a wide range of frequency shifts with respect to the mass and position of an added particle. This model is an intuitive way for designers to develop resonant nanosensors vibrating at large amplitudes for mass detection. Particularly, it is demonstrated that the mass and position of a particle can be determined based on the proposed model reduced to the first bending mode coupled with the analytical expressions of the linear frequency shifts for the second and third modes.

Nonlinear vibrations of clamped-free circular cylindrical shells

Only experimental studies are available on large-amplitude vibrations of clamped-free shells. In the present study, large-amplitude nonlinear vibrations of clamped-free circular cylindrical shell are numerically investigated for the first time. Shells with perfect and imperfect shape are studied. The Sanders–Koiter nonlinear shell theory is used to calculate the elastic strain energy. Shell displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series, i.e. harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. All boundary conditions are satisfied. The system is discretized by using natural modes of the shell and Lagrange equations by an energy approach, retaining damping through Rayleigh's dissipation function. Different expansions involving from 18 to 52 generalized coordinates are used to study the convergence of the solution. The nonlinear equations of motion are numerically studied by using arclength continuation method and bifurcation analysis. Numerical responses to harmonic radial excitation in the spectral neighborhood of the lowest natural frequency are compared with experimental results available in literature. The effect of geometric imperfections and excitation amplitude are numerically investigated and fully explained.

Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors

Large-amplitude non-linear vibrations of micro- and nano-electromechanical resonant sensors around their primary resonance are investigated. A comprehensive multiphysics model based on the Galerkin decomposition method coupled with the averaging method is developed in the case of electrostatically actuated clamped–clamped resonators. The model is purely analytical and includes the main sources of non-linearities as well as fringing field effects. The influence of the higher modes and the validation of the model is demonstrated with respect to the shooting method as well as the harmonic balance coupled with the asymptotic numerical method. This model allows designers to investigate the sensitivity variation of resonant sensors in the non-linear regime with respect to the electrostatic forcing.

Vibrational spectra of Fe3P and Fe2P metal-metalloid compound crystallites: Phonon and breather excitations

The spectra of small-amplitude single-phonon vibrations and large-amplitude nonlinear multiphonon vibrations of Fe3P and Fe2P metal-metalloid compound crystallites are investigated. The vibrational spectra of 3D crystallites of gradually increasing volumes are calculated within a microscopic approach using previously tested interatomic interaction potentials. The calculated energy of the main low-frequency peak is closer to its experimental value in comparison with previous calculations, which reduces the significant discordance between the experimental and theoretical spectra. The fine structure of the high-frequency part of the vibrational spectrum is discussed. A study of large-amplitude nonlinear vibrations (which determine the diffusion of atoms and dispersive processes in materials) showed that a specific nonlinear breather mode of vibration can be generated in the crystallites in question, which is a genetic precursor of nonlinear self-localized vibrations.

Modal Effects on the Local Heat Transfer Characteristics of a Vibrating Body

This is an experimental investigation of the effects of forced transverse vibrations on the local heat transfer characteristics of a heated, pinned-pinned beam. In particular, the response of a cylindrical beam near its first two natural frequencies, corresponding to the first two vibration modes, is considered. The results show that there is a strong spatial variation in the local Nusselt number and that these variations are closely related to the mode shape of the response. Because the heat transfer measurements were taken at the resonance frequencies, where the structural response was greatest, the measured Nusselt numbers provide an upper bound for the increased convection due to flexible body vibrations, i.e., in the absence of any rigid-body mode. The possibility of large-amplitude nonlinear vibrations are discussed (though they were not witnessed experimentally) in a theoretical framework. [S0022-1481(00)01702-3]