An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia

In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal.     

An exact solution for a rotating body with negligible mass

If it becomes possible to test general relativity by laboratory experiments on rotation, the ratios of mass to angular momentum per unit mass are likely to be extremely small. Solutions for a rotating body with low m/a are therefore of interest. Here I discuss Papapetrous exact solution, which has zero mass and arbitrary angular momentum.     

Dynamics of Boolean networks: an exact solution

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.     

Dynamic analysis of Bernoulli-Euler piezoelectric nanobeam with electrostatic force

The effect of electrostatic force on the dynamic response of a Bernoulli-Euler piezoelectric nanobeam is analyzed in this paper.The governing equations with the electrostatic stress are derived based on a variational principle.Static bending problem of simply supported and cantilever beam is considered.The influence of the electrostatic force on the first four natural frequencies is discussed.It is shown that when the beam thickness is small,the effect of the electrostatic force is significant.When the beam thickness is large,the electrostatic force is insignificant and can be neglected.The results also indicate that one can adjust the natural frequency of a nanobeam by applying appropriate voltage.     

A comment on an exact solution with a stiff equation of state

Recently, an analytical stellar model with a stiff equation of state and density distribution= _c (1-r ²/r _o ² ) was presented. We show that such a solution cannot exist.     

The exact solution of an octagonal rectangle-triangle random tiling

We present a detailed calculation of the recently published exact solution of a random tiling model possessing an eightfold-symmetric phase. The solution is obtained using the Bethe Ansatz and provides closed expressions for the entropy and phason elastic constants. Qualitatively, this model has the same features as the square-triangle random tiling model. We use the method of P. Kalugin, who solved the Bethe Ansatz equations for the square-triangle tiling which were found by M. Widom.     

In-plane vibrations of an elastically cantilevered circular arc with a tip mass

Upper and lower bounds are determined for the fundamental frequency of in-plane, transverse vibration of the structural system described in the title in the case of constant cross-section and moment of inertia. The upper bound is determined by approximating the fundamental mode shape with a polynomial co-ordinate function in the angular co-ordinate which includes an exponential optimization parameter. The fundamental frequency equation is generated by means of the Rayleigh-Ritz method and the resulting upper bound is minimized with respect to the previously mentioned exponential parameter. The lower bound for the frequency coefficient is obtained by means of an extension of Dunkerley's method. It is felt that the methodologies developed in the present study are especially useful in the case of variable cross-section of the arch structure, presence of internal supports, etc.     

Physical properties of an exact spherically symmetric solution with shear in general relativity

The properties of an exact spherically symmetric perfect fluid solution obtained in non-comoving coordinates are examined. This solution contains shear, and the pressure and the density are positive in the interior of the fluid. Their respective gradients with respect to comoving radial coordinate are equal and negative, and the speed of sound in this fluid is less than the speed of light in vacuum and is increasing outwards. There is a singularity at the center of the fluid since the pressure and the density become infinite there, though their ratio becomes unity. This singularity is naked, since there does not exist a trapped surface in the fluid outside this singularity. The circumference is an increasing function of radical comoving coordinate, and the mass function is positive and is increasing outwards. There are no tidal forces in radial direction, but the tidal forces normal to this direction are non-vanishing. We also give the kinematic quantities for this fluid. However, it is not possible to match this solution with an exterior vacuum Schwarzschild solution. Moreover, the dominant energy condition produces imaginary values for the sound speed.     

Application of an electrostatically actuated cantilevered carbon nanotube with an attached mass as a bio-mass sensor

In the present paper, another latent capability of SWCNT as a mass sensor is investigated. The relationship between the resonant frequency, dynamic pull-in voltage at the resonance frequency shift, and the attached mass is established by using the nonlocal Euler–Bernoulli beam theory. Using this relationship, a general closed-form nonlinear sensor-equation has been derived for the detection of the mass attached to the SWCNT. The aim of this study and present model is to show the sensitivity of the Cantilevered SWCNT to the values and positions of attached mass. Moreover, the results indicate that by increasing the value of attached mass and considering a single non-local scaling parameter (e₀), the values of dynamic pull-in voltage at the resonance frequency shift are decreased. Because of the small scaling parameter (e₀), the mass sensitivity of carbon nanotube increases, when the position of the attached mass is in the tip of a Cantilevered SWCNT length. The authority and the accuracy of these formulas are examined with other pull-in sensor equations in literatures. The results demonstrate that the new sensor equation can be applied for CNT-based mass sensors with rational accuracy.     

Problem of plane EM wave self-action in multilayer structure: an exact solution

To solve exactly the problem of plane EM wave interaction with multilayer nonlinear structures a non-traditional theoretical method is applied. Wave equation's solution in each layer of a structure is presented in the special form of a single expression. This form of field presentation does not require the fulfillment of superposition principle, contrary to the traditional approach. Multilayer structures of threefold quarter-wavelength bilayers with alternate linear and nonlinear indices are investigated. Multistable dependences of reflection coefficients on incident electric field amplitude are obtained for negative and positive nonlinearity. The results of investigation not only permits to deeply understand the phenomena of intense electromagnetic wave interaction with multilayer structures but also will be very useful for elaboration of nonlinear devices.     

On an exact classical wave solution of the Einstein equations

The J. Scherk and J. H. Schwarz generalization of a Peres solution is shown to be the Peres solution in a different system of coordinates.     

Thin film flow of an Oldroyd 8-constant fluid: An exact solution

This Letter looks at the exact solution for the thin film flow of an Oldroyd 8-constant fluid down a vertical cylinder. The arising nonlinear differential equation is first integrated analytically and then solved numerically. The effects of pertinent fluid parameters on the velocity are seen.     

Low frequency and weakly non-linear conductance of a double-barrier structure: an exact solution

For a double-barrier tunneling system, the low frequency AC admittance and the weakly nonlinear DC conductance can be calculated exactly. We present this solution and compare with that obtained approximately using the Breit-Wigner form of the scattering matrix.     

Vibration of a rotating beam with tip mass

The vibration frequency of a rotating beam with tip mass is investigated. The finite element method is used, a third order polynomial being assumed for the variation of the lateral displacement. The effects of the root radius, the setting angle and the tip mass are incorporated into the finite element model. The results are compared with results from previous authors utilizing Myklestad and extended Galerkin methods. The results show that the setting angle has a significant effect on the first mode frequencies but not on the high frequencies. The tip mass tends to depress the frequencies at low speeds of rotation but it tends to increase the frequencies at high speeds of rotation. The results of this work have applications in wind turbine rotors, helicopter rotors, etc., and the method used here can be extended to investigate the vibration frequency of flexible blade auto cooling fans.