Free vibrations of a thin elastica by normal modes

In this work we investigate the existence, stability and bifurcation of periodic motions in an unforced conservative two degree of freedom system. The system models the nonlinear vibrations of an elastic rod which can undergo both torsional and bending modes. Using a variety of perturbation techniques in conjunction with the computer algebra system MACSYMA, we obtain approximate expressions for a diversity of periodic motions, including nonlinear normal modes, elliptic orbits and non-local modes. The latter motions, which involve both bending and torsional motions in a 2:1 ratio, correspond to behavior previously observed in experiments by Cusumano.     

A nonlinear composite shell theory

A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

Search for and elimination of dependences between vibrational coordinates in modeling spectra of large molecules

We have investigated an algorithm allowing us to reliably identify an arbitrary number of complex linear dependences between vibrational coordinates in a molecular model of very high dimensionality. These dependences are eliminated in the step for diagonalization of the kinetic part of the vibrational Hamiltonian. We have carried out computer experiments allowing us to propose optimal rules for designing appropriate computer programs for working with a vibrational Hamiltonian of very high dimensionality. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 73, No. 5, pp. 561–565, September–October, 2006.     

Stability chart of parametric vibrating systems using energy-rate method

Based on the integral of energy and numerical integration, we introduce, develop, and apply a general algorithm to predict parameters of a parametric equation to produce a periodic response. Using the new method, called energy-rate, we are able to find not only stability chart of a parametric equation which indicates the boundaries of stable and unstable regions, but also periodic responses that are embedded in stable or unstable regions.There are three main important advantages in energy-rate method. It can be applied not only to linear but also to non-linear parametric equations; most of the perturbation methods cannot. It can be applied to large values of parameters; most of the perturbation methods cannot. Depending on the accuracy of numerical integration method, it can also find the value of parameters for a periodic response more accurate than classical methods, no matter if the periodic response is on the boundary of stability and instability or it is a periodic response within the stable or unstable region.In order to introduce the energy-rate method and indicate its advantages we apply the method to the standard Mathieu's equation,

     

Concerning virial-based estimations of strength of bonding intermolecular interactions in molecular crystals and supramolecular complexes

It is shown that the electronic virial-based correlation should be used to estimate bonding contributions to the rigidity of molecular vibrations in crystals.     

Specificity of internal rotation in aldehydes with three-membered rings

The results of anharmonic calculations of low-frequency vibrations of aldehyde molecules containing a three-membered cycle with a π-bond are presented. The theoretical exploration of the conformational behavior and geometrical structure of two related molecules is carried out and potential energy surface sections along the coordinates of internal rotation and non-planar vibration close in frequency are constructed and analyzed. The possibility of kinematic coupling of these two vibrations and the complexity of their forms are especially considered.     

Structure of ScBr3 and Sc2Br6 molecules determined by the synchronous gas-phase electron diffraction and mass spectrometric experiment and quantum chemical calculations

The structure of monomeric and dimeric molecules of scandium tribromide is studied by the synchronous electron diffraction and mass spectrometric experiment at T = 888(10) K and also by the quantum chemical calculations. The experimental data on the structural parameters of ScBr₃ molecule were obtained for the first time; also for the first time the molecular structure of Sc₂Br₆ dimeric molecule was studied. It is found that the ScBr₃ molecule has the C _3v effective configuration with the distance r _g (Sc-Br) = 2.430(3) Å and the valence angle ∠_g(Br-Sc-Br) = 117.6(5)°. The equilibrium structure of the given molecule is planar with D _3h symmetry. According to a theoretical study by DFT and MP2 methods, Sc₂Br₆ molecule has the equilibrium structure of D _2h symmetry with four Sc-Br bridge bonds. It was confirmed by the results of the electron diffraction analysis.