Free vibration of a kinked cantilever with attached masses

In this paper free vibration characteristics of a centrally kinked cantilever beam of unit mass carrying masses at the kink (m(k)) and at the tip (m(t)) are analyzed. Frequency factors are presented for the first two modes for different combinations of m(k),m(t) and the kink angle delta. A relationship of the form f(m(k),m(t), delta) = m(k) + m(t)(4 + 10/3 cos delta+ 2/3 cos(2) delta)=const appears to give the same fundamental frequency for a given delta and different combinations of [m(k), m(t)]. Mode shapes as well as bending moments at the support and at the kink are also discussed. The utility of a discrete beam model in understanding the free vibration characteristics is also highlighted.

Controversy on the Free Energy of Formation of CaO Additional Evidence in Support of Thermochemical Data

The standard free energies of formation of CaO derived from a variety of high-temperature equilibrium measurements made by seven groups of experimentalists are significantly different from those given in the standard compilations of thermodynamic data. Indirect support for the validity of the compiled data comes from new solid-state electrochemical measurements using single-crystal CaF2 and SrF2 as electrolytes. The change in free energy for the following reactions are obtained: CaO + MgF2 --> MgO + CaF2 Delta G degrees = -68,050 -2.47 T(+/-100) J mol(-1) SrO + CaF2 --> SrF2 + CaO Delta G degrees = -35,010 + 6.39 T (+/-80) J mol(-1) The standard free energy changes associated with cell reactions agree with data in standard compilations within +/- 4 kJ mol(-1). The results of this study do not support recent suggestions for a major revision in thermodynamic data for CaO.

Discussion of enthalpy, entropy and free energy of formation of GaN

Presented in this letter is a critical discussion of a recent paper on experimental investigation of the enthalpy, entropy and free energy of formation of gallium nitride (GaN) published in this journal [T.J. Peshek, J.C. Angus, K. Kash, J. Cryst. Growth 311 (2008) 185-189]. It is shown that the experimental technique employed detects neither the equilibrium partial pressure of N-2 corresponding to the equilibrium between Ga and GaN at fixed temperatures nor the equilibrium temperature at constant pressure of N-2. The results of Peshek et al. are discussed in the light of other information on the Gibbs energy of formation available in the literature. Entropy of GaN is derived from heat-capacity measurements. Based on a critical analysis of all thermodynamic information now available, a set of optimized parameters is identified and a table of thermodynamic data for GaN developed from 298.15 to 1400 K.

Some studies on free vibration of composite laminates

Free vibration analysis is carried out to study the vibration characteristics of composite laminates using the modified shear deformation, layered, composite plate theory and employing the Rayleigh-Ritz energy approach. The analysis is presented in a unified form so as to incorporate all different combinations of laminate boundary conditions and with full coverage with regard to the various design parameters of a laminated plate. A parametric study is made using a beam characteristic function as the admissible function for the numerical calculations. The numerical results presented here are for an example case of fully clamped boundary conditions and are compared with previously published results. The effect of parameters, such as the aspect ratio of plates, ply-angle, number of layers and also the thickness ratios of plies in laminates on the frequencies of the laminate, is systematically studied. It is found that for anti-symmetric angle-ply or cross-ply laminates unique numerical values of the thickness ratios exist which improve the vibration characteristics of such laminates. Numerical values of the non-dimensional frequencies and nodal patterns, using the thickness ratio distribution of the plies, are then obtained for clamped laminates, fabricated out of various commonly used composite materials, and are presented in the form of the design curves.

Axisymmetric free vibration of thick annular plates

Free vibrationo f annularp latesh asb eens tudiedu singt he methodo f initial functionsN. umericalr esultso f the naturalf requencieasr e obtainedfo r two typicals upportc onditionsT. he presenrt esultsd epartf rom the classicapll atet heoryf or higherm odesa s wella sf or thickerp latesb, ut is in gooda greemenwt ith Mindlin's improve,pdl ateth eory.

Free vibrations of a reservoir-foundation system

The natural frequencies of a reservoir-foundation system are calculated by treating the foundation as a system of linear springs with inertia. The reservoir is treated as consisting of compressible liquid, and the influence of waves at the free surface is included. It is shown that the natural frequencies decrease monotonically as the depth of foundation participating in the motion increases. The influence of waves at the reservoir surface is negligible for the cases normally occurring in practice. It is also shown that the wavelength of motion along the reservoir has no influence on the frequencies when the foundation depth is large compared to the reservoir depth.

Free vibration of rectangular plates of arbitrary thickness

Free vibration of thick rectangular plates is investigated by using the “method of initial functions” proposed by Vlasov. The governing equations are derived from the three-dimensional elastodynamic equations. They are obtained in the form of series and theories of any desired order can be constructed by deleting higher terms in the infinite order differential equations. The numerical results are compared with those of classical, Mindlin, and Lee and Reismann solutions.

Free vibration of rectangular plates of arbitrary thickness with one or more edges clamped

Frequencies of free vibration of rectangular plates of arbitrary thickness, with different support conditions, are calculated by using the Method of Initial Functions (MIF), proposed by Vlasov. Sixth and fourth order MIF theories are used for the solution. Numerical results are presented for three square plates for three thickness ratios. The support conditions considered are (i) three sides simply supported and one side clamped, (ii) two opposite sides simply supported and the other two sides clamped and (iii) all sides clamped. It is found that the results produced by the MIF method are in fair agreement with those obtained by using other methods. The classical theory gives overestimates of the frequencies and the departures from the MIF results increase for higher modes and larger thickness ratios.

Force-free equilibria of magnetized jets

Force-free equilibrium configurations of magnetic-pressure-dominated magnetized supersonic jets confined by slowly varying external pressure are investigated analytically. For the case where internal dissipation mechanisms are active, the lowest-energy field configuration is found to be the superposition of an axisymmetric mode and a helical mode with a wavelength equal to 5 times the jet radius, and the pressure below which the nonaxisymmetric mode becomes energetically favorable is given as 2700 times the product of the 4th power of the magnetic helicity per unit length and the -6th power of the magnetic flux. A model of the total and polarized emission of such a configuration is developed and applied to the extended well-collimated astronomically resolved jet NGC 6251. The model is shown to reproduce significant features such as transverse oscillations of the ridge line, width oscillations and emission knots, the projected magnetic-field configuration, oscillations of the degree of polarization, and the distribution of the Faraday rotation measure.

Free vibrations of non-linear cubic spring mass systems in the presence of coulomb damping

An exact solution for the free vibration problem of non-linear cubic spring mass system with Coulomb damping is obtained during each half cycle, in terms of elliptic functions. An expression for the half cycle duration as a function of the mean amplitude during the half cycle is derived in terms of complete elliptic integrals of the first kind. An approximate solution based on a direct linearization method is developed alongside this method, and excellent agreement is obtained between the results gained by this method and the exact results. © 1970 Academic Press Inc. (London) Limited.

An exact analysis of free vibrations of simply-supported viscoelastic plates

In this paper an exact three-dimensional analysis for free vibrations of a class of simply-supported viscoelastic rectangular plates is given. The characteristic equation defining the eigenvalues is of closed form. Some numerical results are presented for standard linear solids. Results from thin plate and Mindlin theories are also given for the purpose of comparison.

Free energy of mixing of divalent basic oxides with silica

An expression derived for the free energy of mixing of a divalent basic oxide (MO) with SiO2 based on a model of silicate structure, takes into account the distribution of O2- (from MO) into the silica network, the mixing of silicate ions with O2- and the enthalpy of mixing. The resulting expression is ΔGmix=RT{N11n (2N1-N)2/4N1(1-N)+N21n N 2-N/1-N}, where N={(β+N1)-√(β+N 1)2-8βN1N2}/2β β=characteristic constant for the system N1=mol fraction of silica N2=mol fraction of MO. For the proper choice of β, calculated values of the activity of MO for the system PbO-SiO2, MnO-SiO2, FeO-SiO2 and CaO-SiO2 are in good agreement with experiment. The model predicts that the activity of the basic oxide decreases with increase in temperature.

Free vibration of circular plates of arbitrary thickness

Free vibration of circular plates of arbitrary thickness is investigated using the method of initial functions. State-space approach is used to derive the governing equations of the above method. The formulation is such that theories of any desired order can be obtained by deleting higher terms in the infinite-order differential equations. Numerical results are obtained for flexural and extensional vibration of circular plates. Results are also computed using Mindlin's theory and they are in agreement with the present analysis.

Enantioselective and Protecting Group-Free Synthesis of 1-Deoxythionojirimycin, 1-Deoxythiomannojirimycin, and 1-Deoxythiotalonojirimycin

1-Deoxythioglyconojirimycins were synthesized by using a protecting group-free strategy, starting from readily available carbohydrates, in good overall yield. Use of benzyl-triethylammonium tetrathiomolybdate, BnEt3N](2)MoS4, as a sulfur transfer reagent and borohydride exchange resin (BER) reduction of a lactone enabled the efficient synthesis of the title compounds.

Free vibration of a stochastic beam-column using stochastic fem

The free vibrational characteristics of a beam-column, which is having randomly varying Young's modulus and mass density and subjected to randomly distributed axial loading is analysed. The material property fluctuations and axial loadings are considered to constitute independent one-dimensional, uni-variate, homogeneous real, spatially distributed stochastic fields. Hamilton's principle is used to formulate the problem using stochastic FEM. Vibration frequencies and mode shapes are analysed for their statistical descriptions. A numerical example is shown.