Application of a continuum theory to vertical vibrations of a layer of granular material

Many interesting phenomena have been observed in layers of granular materials subjected to vertical oscillations; these include the formation of a variety of standing wave patterns, and the occurrence of isolated features called oscillons, which alternately form conical heaps and craters oscillating at one-half of the forcing frequency. No continuum-based explanation of these phenomena has previously been proposed. We apply a continuum theory, termed the double-shearing theory, which has had success in analyzing various problems in the flow of granular materials, to the problem of a layer of granular material on a vertically vibrating rigid base undergoing vertical oscillations in plane strain. There exists a trivial solution in which the layer moves as a rigid body. By investigating linear perturbations of this solution, we find that at certain amplitudes and frequencies this trivial solution can bifurcate. The time dependence of the perturbed solution is governed by Mathieu’s equation, which allows stable, unstable and periodic solutions, and the observed period-doubling behaviour. Several solutions for the spatial velocity distribution are obtained; these include one in which the surface undergoes vertical velocities that have sinusoidal dependence on the horizontal space dimension, which corresponds to the formation of striped standing waves, and is one of the observed patterns. An alternative continuum theory of granular material mechanics, in which the principal axes of stress and rate-of-deformation are coincident, is shown to be incapable of giving rise to similar instabilities.

Confirmation of the assignment of vibrations of goethite: An ATR and IES study of goethite structure

Transmission electron microscopy (TEM), field emission scanning electron microscopy (FE-SEM/EDS) and X-ray diffraction (XRD) were used to characterize the morphology of synthetic goethite. The behavior of the hydroxyl/water molecular units of goethite and its thermally treated products were characterized using Fourier transform-infrared emission spectroscopy (FT-IES) and attenuated total reflectance–Fourier transform infrared (ATR–FTIR) spectroscopy. The results showed that all the expected vibrational bands between 4000 and 650 cm−1 including the resolved bands (3800–2200 cm−1) were confirmed. A band attributed to a new type of hydroxyl unit was found at 3708 cm−1 and assigned to the FeO–H stretching vibration without hydrogen bonding. This hydroxyl unit was retained up to the thermal treatment temperature of 500 °C. On the whole, seven kinds of hydroxyl units, involving three surface hydroxyls, a bulk hydroxyl, a FeO–H without hydrogen bonding, a nonstoichiometric hydroxyl and a reversed hydroxyl were observed, and three kinds of adsorbed water were found in/on goethite.

Hydraulic instability onset detection in Kaplan turbines by monitoring shaft vibrations

Design of hydraulic turbines has often to deal with hydraulic instability. It is well-known that Francis and Kaplan types present hydraulic instability in their design power range. Even if modern CFD tools may help to define these dangerous operating conditions and optimize runner design, hydraulic instabilities may fortuitously arise during the turbine life and should be timely detected in order to assure a long-lasting operating life. In a previous paper, the authors have considered the phenomenon of helical vortex rope, which happens at low flow rates when a swirling flow, in the draft tube conical inlet, occupies a large portion of the inlet. In this condition, a strong helical vortex rope appears. The vortex rope causes mechanical effects on the runner, on the whole turbine and on the draft tube, which may eventually produce severe damages on the turbine unit and whose most evident symptoms are vibrations. The authors have already shown that vibration analysis is suitable for detecting vortex rope onset, thanks to an experimental test campaign performed during the commissioning of a 23 MW Kaplan hydraulic turbine unit. In this paper, the authors propose a sophisticated data driven approach to detect vortex rope onset at different power load, based on the analysis of the vibration signals in the order domain and introducing the so-called "residual order spectrogram", i.e. an order-rotation representation of the vibration signal. Some experimental test runs are presented and the possibility to detect instability onset, especially in real-time, is discussed.

Spectroscopic vibrations of austinite (CaZnAsO4⋅OH) and its mineral structure : Implications for identification of secondary arsenic-containing mineral

Austinite (CaZnAsO4⋅OH) is a unique secondary mineral in arsenic-contaminated mine wastes. The infrared and Raman spectroscopies were used to characterize the austenite vibrations. The IR bands at 369, 790 and 416 cm−1 are assigned to the ν2, ν3 and ν4 vibrations of AsO43− unit, respectively. The Raman bands at 814, 779 and 403 cm−1 correspond to the ν1, ν3 and ν4 vibrations of AsO43− unit respectively. The sharp bands at 3265 cm−1 for IR and 3270 cm−1 both reveals that the structural hydroxyl units exist in the austenite structure. The IR and Raman spectra both show that some SO4 units isomorphically replace AsO4 in austinite. X-ray single crystal diffraction provides the arrangement of each atom in the mineral structure, and also confirms that the conclusions made from the vibrational spectra. Micro-powder diffraction was used to confirm our mineral identification due to the small quantity of the austenite crystals.

Infinite Dimensional Slow Modulations In A Delayed Model For Orthogonal Cutting Vibrations

We apply the method of multiple scales (MMS) to a well known model of regenerative cutting vibrations in the large delay regime. By ``large'' we mean the delay is much larger than the time scale of typical cutting tool oscillations. The MMS upto second order for such systems has been developed recently, and is applied here to study tool dynamics in the large delay regime. The second order analysis is found to be much more accurate than first order analysis. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy. The main advantage of the present analysis is that infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space. Lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS. Finally, the strong sensitivity of the dynamics to small changes in parameter values is seen clearly.

Assignment of fundamental vibrations of. ,N'-dimethylthiourea,

The infrared spectra of symmetric N,N′-dimethylthiourea (s-DMTU) and its N-deuterated (s-DMTU-d2) species have been measured. The fundamental frequencies have been assigned by comparison with the assignments in structurally related molecules and the infrared band shifts on N-deuteration, S-methylation, available Raman data and with the aid of theoretical band assignments from normal coordinate treatments for s-DMTU-d0 and -d2. A force field is derived for s-DMTU by transferring the force constants chiefly from N-methylthiourea and the subsequent refinement of the force constants by a least squares procedure.

Non-linear flapping vibrations of rotating blades

The non-linear equations of motion of a rotating blade undergoing extensional and flapwise bending vibration are derived, including non-linearities up to O (ε3). The strain-displacement relationship derived is compared with expressions derived by earlier investigators and the errors and the approximations made in some of those are brought out. The equations of motion are solved under the inextensionality condition to obtain the influence of the amplitude on the fundamental flapwise natural frequency of the rotating blade. It is found that large finite amplitudes have a softening effect on the flapwise frequency and that this influence becomes stronger at higher speeds of rotation.

Vibrations of a Beam in Variable Contact With a Flat Surface

We study small vibrations of cantilever beams contacting a rigid surface. We study two cases: the first is a beam that sags onto the ground due to gravity, and the second is a beam that sticks to the ground through reversible adhesion. In both cases, the noncontacting length varies dynamically. We first obtain the governing equations and boundary conditions, including a transversality condition involving an end moment, using Hamilton's principle. Rescaling the variable length to a constant value, we obtain partial differential equations with time varying coefficients, which, upon linearization, give the natural frequencies of vibration. The natural frequencies for the first case (gravity without adhesion) match that of a clamped-clamped beam of the same nominal length; frequencies for the second case, however, show no such match. We develop simple, if atypical, single degree of freedom approximations for the first modes of these two systems, which provide insights into the role of the static deflection profile, as well as the end moment condition, in determining the first natural frequencies of these systems. Finally, we consider small transverse sinusoidal forcing of the first case and find that the governing equation contains both parametric and external forcing terms. For forcing at resonance, w find that either the internal or the external forcing may dominate.

Molecular vibrations, electronic structure and conformation of diprotonated thiocarbohydrazide

The infrared spectra of diprotonated species of thiocarbohydrazide and its perdeuterated derivative have been examined in the crystalline state. A complete vibrational assignment with a full normal coordinate treatment based on a Urey—Bradley type intramolecular potential Function supplemented with a valence force function for the out of plane and torsional modes is proposed and the origin of the amide II band splittings is explained. A CNDO/2 study of diprotonated thiocarbohydrazide and its neutral molecule is undertaken and the changes in the molecular electronic structures and conformations consequent to protonation are determined and briefly discussed. The magnitude of the N—N+H3 torsional barrier is estimated to be 21 kJ mol− (5.0 kcal mol−1) whereas the barrier for the C—N group is found to be 92 kJ mol−1 (22.0 kcal mol−1).

Non-linear flapping vibrations of rotating blades

The non-linear equations of motion of a rotating blade undergoing extensional and flapwise bending vibration are derived, including non-linearities up to O (ε3). The strain-displacement relationship derived is compared with expressions derived by earlier investigators and the errors and the approximations made in some of those are brought out. The equations of motion are solved under the inextensionality condition to obtain the influence of the amplitude on the fundamental flapwise natural frequency of the rotating blade. It is found that large finite amplitudes have a softening effect on the flapwise frequency and that this influence becomes stronger at higher speeds of rotation.

On the axial vibrations of rotating bars

One of the important developments in rotary wing aeroelasticity in the recent past has been the growing awareness and acceptance of the fact that the problem is inherently non-linear and that correct treatment of aeroelastic problems requires the development of a consistent mathematical model [l]. This has led to a number of studies devoted to the derivation of a consistent set of “second order” non-linear equations, for example, those of Hodges and Dowel1 [2], of Rosen and Friedmann [3], and of Kvaternik, White and Kaza [4], each of which differs from the others on the question of the inclusion of certain terms in the equations of motion. The final form of the equations depends first upon the ordering scheme used for characterizing the displacements and upon the consistency with which this is applied in omitting terms of lower order. The ideal way of achieving this would be to derive the equations of motion with all the terms first included regardless of their relative orders of magnitude and then to apply the ordering scheme.

Transverse vibrations of hybrid laminated plates

In this paper, an attempt is made to obtain the free vibration response of hybrid, laminated rectangular and skew plates. The Galerkin technique is employed to obtain an approximate solution of the governing differential equations. It is found that this technique is well suited for the study of such problems. Results are presented in a graphical form for plates with one pair of opposite edges simply supported and the other two edges clamped. The method is quite general and can be applied to any other boundary conditions.

Vibrational assignments of five-membered heterocyclic compounds. Normal vibrations of oxazolidine-2-one and -2-thione

Infrared spectra of oxazolidine-2-one (Oxo), -2-thione (Oxt) and their N-deuteriated derivatives have been measured over the range 4000-20 cm−1. The fundamental frequencies of these molecules have been assigned on the basis of normal coordinate calculations carried out using a Urey-Bradley potential function supplemented with valence type constants for the out-of-plane modes of the planar skeleton. The results of the vibrational analyses are discussed and correlated with the assignments available for the other related five membered heterocyclic molecules.

Vibrations of laminated beams

Governing equations in the form of simultaneous ordinary differential equations have been derived for natural vibration analysis of isotropic laminated beams. This formulation includes significant secondary effects such as transverse shear and rotatory inetia. Through a numerical example, the influence of these secondary effects has been studied.