Chaotic vibrations of size-dependent flexible rectangular plates.

A mathematical model describing nonlinear vibrations of size-dependent rectangular plates is proposed. The plates are treated as the Cosserat continuum with bounded rotations of their particles (pseudo-continuum). The governing partial differential equations (PDEs) and boundary/initial conditions are obtained using the von Kármán geometric relations, and they are yielded by the energetic Hamilton principle. The derived mixed-form PDEs are reduced to ordinary differential equations and algebraic equations (AEs) using (i) the Galerkin-Krylov-Bogoliubov method (GKBM) in higher approximations, and then they are solved with the help of a combination of the Runge-Kutta methods of the second and fourth order, (ii) the finite difference method (FDM), and (iii) the Newmark method. The convergence of FDM vs the interval of the space coordinate grids and of GKBM vs the number of employed terms of the approximating function is investigated. The latter approach allows for achieving reliable results by taking account of almost infinite-degree-of-freedom approximation to the regular and chaotic dynamics of the studied plates. The problem of stability loss of the size-dependent plates under harmonic load is also tackled.

On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness.

We construct a mathematical model of non-linear vibration of a beam nanostructure with low shear stiffness subjected to uniformly distributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elastic material obey the Hooke law and are governed by the kinematic third-order approximation (Sheremetev-Pelekh-Reddy model). The von Kármán geometric non-linear relation between deformations and displacements is taken into account. In order to describe the size-dependent coefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations, as well as the initial and boundary conditions. A solution to the dynamical problem is found via the finite difference method of the second order of accuracy, and next via the Runge-Kutta method of orders from two to eight, as well as the Newmark method. Investigations of the non-linear nanobeam vibrations are carried out with a help of signals (time histories), phase portraits, as well as through the Fourier and wavelet-based analyses. The strength of the nanobeam chaotic vibrations is quantified through the Lyapunov exponents computed based on the Sano-Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling procedure allowed us to achieve reliable results. In particular, we have detected chaotic and hyper-chaotic vibrations of the studied nanobeam, and our results are authentic, reliable, and accurate.

Nonlinear dynamics of contact interaction of a size-dependent plate supported by a size-dependent beam.

A mathematical model of complex vibrations exhibited by contact dynamics of size-dependent beam-plate constructions was derived by taking the account of constraints between these structural members. The governing equations were yielded by variational principles based on the moment theory of elasticity. The centre of the investigated plate was supported by a beam. The plate and the beam satisfied the Kirchhoff/Euler-Bernoulli hypotheses. The derived partial differential equations (PDEs) were reduced to the Cauchy problems by the Faedo-Galerkin method in higher approximations, whereas the Cauchy problem was solved using a few Runge-Kutta methods. Reliability of results was validated by comparing the solutions obtained by qualitatively different methods. Complex vibrations were investigated with the help of methods of nonlinear dynamics such as vibration signals, phase portraits, Fourier power spectra, wavelet analysis, and estimation of the largest Lyapunov exponents based on the Rosenstein, Kantz, and Wolf methods. The effect of size-dependent parameters of the beam and plate on their contact interaction was investigated. It was detected and illustrated that the first contact between the size-dependent structural members implies chaotic vibrations. In addition, problems of chaotic synchronization between a nanoplate and a nanobeam were addressed.

On the iterative methods of linearization, decrease of order and dimension of the Karman-type PDEs.

Iterative methods to achieve a suitable linearization as well as a decrease of the order and dimension of nonlinear partial differential equations of the eighth order into the biharmonic and Poisson-type differential equations with their simultaneous linearization are proposed in this work. Validity and reliability of the obtained results are discussed using computer programs developed by the authors.

Chaotic dynamics of flexible Euler-Bernoulli beams.

Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c(2)) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q(0) and frequency ω(p) of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.

Immediate stiffness of the C5-C6 segment after discectomy with the Cloward technique: an in vitro biomechanical study on a human cadaveric model.

OBJECTIVE: To determine whether the Cloward technique of cervical discectomy and fusion increases immediate postoperative stiffness of single cervical motion segment after application of interbody dowel bone graft. METHODS: We measured and compared the stiffness of single-motion segments in cadaveric cervical spines before and immediately after interbody fusion with the Cloward technique. Changes in range of motion and stiffness of the C5-C6 segment were measured in a bending flexibility test (flexion, extension, lateral bending and axial rotation) before and after a Cloward procedure in 11 fresh-frozen human cadaveric specimens from the 4th through the 7th vertebrae. RESULTS: The Cloward procedure produced a statistically significant increase in stiffness of the operated segment in flexion and lateral bending when compared with the intact spine. The less stiff the segment before the operation, the greater the increase in its postoperative flexural stiffness (statistically significant). The Cloward procedure produced nonuniform changes in rotational and extensional stiffness that increased in some specimens and decreased in others. CONCLUSION: Our data demonstrate that Cloward interbody fusion increases immediate postoperative stiffness of an operated segment only in flexion and lateral bending in cadaveric specimens in an in vitro environment. Thus, Cloward fusion seems a relatively ineffective method for increasing the stiffness of a construct. This may add to discussion on the use of spinal instrumentation and postoperative management of patients after cervical discectomy, which varies from bracing in hard collars through immobilization in soft collars to no external orthosis.