Shear and rotatory inertia effect on Beck's column

In this paper the influence of transverse shear deformation and rotatory inertia upon the flutter load of Beck's column with various support characteristics for a variety of slenderness ratios and cross-sectional shapes is presented. The analysis is based on Cowper's formulae for establishing Timoshenko's shear coefficient K'. From this investigation it is found that the inclusion of these parameters may have an appreciable destabilizing effect in the case of a fully fixed cantilever, and particularly in the case of a partially fixed cantilever with an attached mass at the support. This occurs especially in columns with low critical slenderness ratios and thin cross-sections. Moreover, it is noticed that the flutter frequency— for flutter loads obtained by coalescing either of the first and second or second and third flexural eigenfrequencies-never exceeds the precise value 11·01l… of Beck's column.     

Influence of a viscoelastic foundation on the stability of Beck's column: An exact analysis

The stability of Beck's column supported by three different viscoelastic foundations, viz., the standard linear solid, the Maxwell and the Kelvin-Voigt, is investigated. Closed form stability criteria are obtained for the entire range of system parameters through an exact dynamic analysis for each foundation model. The results for the Kelvin-Voigt model show that for a given stiffness parameter of the foundation the critical load increases with an increase in damping and reaches a limiting value for large damping. Unlike the case of conservative loading, the Maxwell foundation is shown to have a positive influence on the stability of this non-conservative problem. Furthermore, for this model, an optimum combination of foundation parameters exists to yield the maximum flutter load. The standard linear solid foundation combines the characteristics of Maxwell and Kelvin-Voigt models, as expected.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

Flexural-flexural oscillations of Beck's column subjected to a planar harmonic excitation

The non-linearly coupled flexural-flexural oscillations of a non-conservative column with two independent load parameters are investigated by analytical methods with the objective of determining the quantitative and the qualitative behavior of the response. The column is subjected to a planar periodic distributed load whose frequency is near the eigenfrequency of its free bending oscillations. The analytical results of the analysis are compared with those obtained by numerical integration of a set of non-linear differential equations obtained by the application of Galerkin's method to the original equations.     

The influence of rotatory inertia,tip mass,and damping on the stability of a cantilever beam on an elastic foundation

The stability of a uniform viscoelastic cantilever resting on an elastic foundation, carrying a tip mass, and subjected to a follower force at its free end is investigated. The effects of the rotatory inertia of the beam, the transverse and rotatory inertias of the tip mass, and the foundation modulus, which characterizes a Winkler type of elastic foundation, are included in the partial differential equation of motion and boundary conditions, and the influence of these quantities on the value of the critical flutter load parameter Q_f is sought. The exact forms of the fundamental frequency equations are derived for the cases of a viscoelastic and a purely elastic beam, and these equations are solved numerically for Q_f These numerical results reveal that Q_f depends strongly upon the foundation modulus for the cantilever carrying a tip mass or possessing rather small internal damping. In the absence of damping and a tip mass, the value of Q_f, computed upon the inclusion of the rotatory inertia of the beam in the formulation of the equation of motion, is decreased slightly and continues to decrease in essentially a linear manner as the value of the foundation modulus parameter κ is decreased. Moreover, when the effect of very small internal damping is included, the value of Q_f computed when the rotatory inertia of the beam is neglected increases slowly in an essentially linear fashion as x increases, whereas, when the effect of rotatory inertia is retained, the value of Q_f decreases as κ is increased. Additional numerical results are reported graphically.     

A useful formula for Witten's mass hierarchy

A general formula is derived for the effective potential to be used in Witten's mass hierarchy.     

The Miura transformations of Kaup's equation and of Mikhailov's equation

Two types of Miura transformations have been found. One relates Kaup's equation to Sawada-Kotera's equation and another relates the generalized Mikhailov equation to a model equation for a shallow water wave.     

The quark mass ratio and the η' mass

Within a class of null-plane field theories we show that the Gell-Mann Oakes, Renner value for the quark mass ratio is not inconsistent with the η' mass.     

Validity of the Talmi's mass formula for single closed shell nuclei

A quadratic mass formula proposed by Talmi for identical nucleons in an open and non degenerate shell is discussed. Corrections coming from the breaking of generalized seniority are examined.     

A possible unification of Newton's and Coulomb's forces

We have considered electric charge as the fourth component of the particle momentum in five-dimensional space–time. The fifth dimension has been compactified on a circle with an extremely small radius determined from the fundamental physics constants. First, we have given equations in the framework of five-dimensional special relativity and determined the corresponding reduction to four-dimensional space–time. Then, in order to obtain an appropriate charge-to-mass ratio and to avoid the Fourier modes problem, we have considered the propagation of an off-mass shell particle in the five-dimensional space–time which can be interpreted as the motion of an on-mass shell particle in the four-dimensional world we experience. As an example, we have discussed the five-dimensional kinematic equations associated with the electron-positron annihilation process into two photons. Finally, the consequences on the gravitational interaction between two elementary charged particles has been studied. As a main result, we have obtained a unification of Newton's gravitational and Coulomb's electrostatic forces.     

The vacuum violates Bell's inequalities

It is found that the vacuum state in any Bose or Fermi free quantum field theory violates Bell's inequalities maximally, i.e. in principle, with suitable detectors maximal violations of Bell's inequalities may be obtained without setting up a source. We explain, however, why it would be difficult to measure such violations.     

The effect of the earth's magnetic field on auger analysis

Semi-quantitative analyses of thin films or surfaces are commonly obtained from the peak intensities in the differentiated Auger spectrum. To reduce effects of surface roughness, beam focus and electron current, ratios of peak heights are used rather than absolute values. In performing analysis of CdSe single crystal and thin film samples in a commercial Auger analyzer fitted with a cylindrical mirror analyzer (CMA), the ratio of the Cd(376 eV) to Se(1315 eV) peaks was found to vary by as much as 15% when the diameter of the incident electron beam was increased from 5 to 60 μm. The effect was found to be due to an energy-dependent shift of the electron beam caused by the earth's magnetic field. The electron transmission of the CMA was measured as a function of the primary electron beam spot position on the sample. The transmission decreases rapidly once the spot falls outside an area with a radius ～25 μm. Due to this response, the relative shift in position caused by the magnetic field produces variations in peak ratios when the spot size is changed. This effect will produce inaccurate analysis if the Auger peaks differ significantly in energy and the primary electron beam spot size is large, and accounts for the observed 15% variation in $\text{Cd}\text{Se}$ ratio.     

The tensor part of Skyrme's interaction

A tensor part is added to central and spin-orbit parts of Skyrme's effective interaction and its contribution to the Hartree-Fock spin-orbit coupling of spherical spin-unsaturated nuclei is analysed. Minor improvements to spin-orbit splittings can be achieved if the strengths of the tensor interaction are treated as free parameters, but the fit to experiment deteriorates if the tensor interaction contribution is estimated for a realistic interaction.     

The influence of the geometry of the tip on the STM images

The theoretical studies presented in this paper concern the influence of the atomic structure of the tip on the tunneling of electrons between the tip and the sample in STM. This problem has been discussed for the system formed by Al(001) surface and aluminium tip, where different geometries of the tip were taken into consideration. The obtained results have shown how various geometries of the tip induce considerably different STM images of the surface.