Weighted linearization technique for period approximation in large amplitude non-linear oscillations

Period approximations for conservative, one-dimensional, non-linear oscillating systems are considered. The approximation technique used is linearization of the non-linear restoring function governing the system. While in previous papers the best period approximations in the small amplitude limit have been discussed, the best large amplitude results are investigated herein, with use of linearization based on weight functions of the power type. The non-linear functions examined here are odd polynomials, sine, hyperbolic sine, tangent, and hyperbolic tangent.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

The electrical breakdown of gases in the presence of crossed electric and magnetic fields

The breakdown characteristics of a gas in the presence of crossed electric and magnetic fields are discussed in terms of the Townsend ionization coefficients. The “equivalent pressure” concept is used to assess the effect of a transverse magnetic field on the first Townsend coefficient and the objections which have been raised to the application of this approach to breakdown potentials are shown to be removed by a consideration of the dependence of the second Townsend coefficient upon electric and magnetic field strengths.     

On the connection between the inverse transform method and the exact quantum eigenstates

The “inverse scattering transformation”, which has been used to solve certain non-linear field theories classically, is discussed in the context of the quantized version of these theories. In particular we consider the non-linear Schrödinger equation and the massive Thirring model. We find that certain Jost functions of the associated scattering problem lead already, in quantizing the theory, to creation operators for the exact eigenstates of the corresponding Hamiltonians.     

Imaging properties of electrostatic energy analyzers with toroidal fields

Numerical values of the second-order aberration coefficients are given for two classes of toroidal fields, namely (1) the “quasi-spherical” field with c = 1, c′ ≠ ? 1, and (2) fields with c in the region 1.6–1.8. For the “quasi-spherical” fields, the second-order coefficients are given as functions of the deflection angles ω for 90° ≤ ω ≤ 180°. The third-order coefficients for the spherical field are given in the same region.For the devices in the second class, which have one intermediary axial image, the second-order coefficients are studied as functions of c and ω, subject to the condition that double-focussing should be obtained. The conditions under which the radial and axial image planes coincide are considered. The influence of axially curved entrance and exit surfaces is investigated. One special case (ω = 180°, c = 1.691, c′= ? 1.11), which may be favourable for an electron monochromator is discussed.     

Statistical mechanics of linear molecules II: Density expansions,intermolecular potentials and virial coefficients

In the Percus-Yevick and convolution-hypernetted-chain equations obtained in the previous paper, a density expansion for the correlation functions is introduced. To first order in the density, the so-obtained equations are identical and exact. By solving these, the pair correlation functions for linear molecules are obtained explicitly to first order in the density and for arbitrary order in the potential perturbation expansion. From these, the second and third virial coefficients can be extracted for all orders. A generalized charge is defined and used to give generalized multipole expansions for the intermolecular potential. Explicit expressions for this potential model are given up to fourth order. It is shown how the correlation functions, and second and third virial coefficients can be obtained to fourth order for any intermolecular potential with the same perturbation structure.     

Second order front tracking for the Euler equations

A second order front tracking method is developed for solving the hyperbolic system of Euler equations of inviscid fluid dynamics numerically. Meshless front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here second order convergence is achieved by deriving a piecewise linear reconstruction of the piecewise constant front tracking solution. The linearization is performed by decomposing the front tracking solution into its wave components and by linearizing the wave solutions separately. In order to construct a physically correct linearization, the physical phenomena of the front are taken into account in terms of the front types of the previously developed improved front interaction model. This front interaction model is also extended to include front numbers used in the wave decomposition. It is illustrated numerically for Sod’s Riemann problem, the two interacting blast waves problem, and a two-dimensional supersonic airfoil flow validation study that the proposed front tracking method achieves second order convergence also in the presence of strong discontinuities and their interactions.     

Molecular beam surface ionization detection: I. Absolute ionization coefficients and work function of “low work function” Pt(8%W) surfaces

Absolute ionization coefficients β have been measured for K on a “low work function” PtW surface by the field reversal method. The modified Langmuir-Saha equation has been fitted by a least-squares procedure to the static surface ionization values as a function of temperature, and ionization coefficients have been computed from the best fit parameter values. The agreement between the two methods is satisfactory at low temperatures, within 9% at β = 0.7–0.8, with the largest error probably lying in the coefficients from static ionization. By a comparison with other methods used for determinations of β < 1 the field reversal method is shown to be as accurate as the apparently most reliable method used for such work, the microbalance method, first used by Schroen 1963. The Richardson plots for the probably graphite coated metal surface are curved. It is shown, that the curvature does not depend on the choice of the spectral emissivity or other possible experimental errors. The only explanation for the curved plots, which appears to be consistent with the experiments, is a true non-linear temperature dependence of the work function, probably due to the semiconducting surface layer. Non-linear leastsquares fits have been used to fit the theoretical expressions to the electron emission current, the surface ionization current and also to a combined quantity, in which the temperature dependence of the work function has been effectively removed. The nature of the PtW surface and some earlier, inconsistent investigations of this surface are also discussed.     

Mean square response of a duffing oscillator to a modulated white noise excitation by the generalized method of equivalent linearization

The non-stationary random response of non-linear systems is considered. The technique of equivalent linearization is generalized for application to non-stationary non-linear random systems and several approximate methods of solution are presented. The example of a Duffing oscillator is studied in detail and its mean square response is evaluated and discussed.     

Multiple-scaling methods for Monte Carlo simulations of radiative transfer in cloudy atmosphere

Two multiple-scaling methods for Monte Carlo simulations were derived from integral radiative transfer equation for calculating radiance in cloudy atmosphere accurately and rapidly. The first one is to truncate sharp forward peaks of phase functions for each order of scattering adaptively. The truncated functions for forward peaks are approximated as quadratic functions; only one prescribed parameter is used to set maximum truncation fraction for various phase functions. The second one is to increase extinction coefficients in optically thin regions for each order scattering adaptively, which could enhance the collision chance adaptively in the regions where samples are rare. Several one-dimensional and three-dimensional cloud fields were selected to validate the methods. The numerical results demonstrate that the bias errors were below 0.2% for almost all directions except for glory direction (less than 0.4%) and the higher numerical efficiency could be achieved when quadratic functions were used. The second method could decrease radiance noise to 0.60% for cumulus and accelerate convergence in optically thin regions. In general, the main advantage of the proposed methods is that we could modify the atmospheric optical quantities adaptively for each order of scattering and sample important contribution according to the specific atmospheric conditions.     

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function

A general theory of monodromy preserving deformation is developed for a system of linear ordinary differential equations $\text{d}\text{Y}\text{d}\text{x}\text{=A(x)Y}$, where A(x) is a rational matrix. The non-linear deformation equations are derived and their complete integrability is proved. An explicit formula is found for a 1-form ω, expressed rationally in terms of the coefficients of A(x), that has the property dω=0 for each solution of the deformation equations. Examples corresponding to the “soliton” and “rational” solutions are discussed.     

Stability of forced periodic response in third order non-linear systems

Conditions for the stability of forced periodic response in third order non-linear systems are obtained after linearization. These conditions are consistent with results obtained by other methods.     

On spectral methods for solving nonlinear acoustics equations

Two very efficient methods for obtaining approximate solutions to nonlinear acoustics equations are discussed. I proposed these methods earlier, but they are still little known. The first method is based on expanding an unknown function into a Taylor series with respect to the coordinate (evolution variable) and on approximate summation of the terms of this series in all orders up to the infinite order. This series can be summed completely only in particular cases, e.g., for a simple wave. It has been noted that the partial summation technique is implemented more easily if all the terms of the series are represented as corresponding topological diagrams. The second method is based on introducing a “nonlinear” phase delay (proportional to the wave amplitude) for the temporal variable in linear solutions of the problem. The application technique of these methods is illustrated by obtaining approximate solutions of the Burgers equation.     

Band model analysis of laboratory methane absorption spectra from 4500 to 10500 Å

Molecular band models are used to derive absorption and pressure coefficients for the methane absorption spectrum from 4500 to 10500 Å at intervals of 10 Å. These coefficients provide a necessary basis for the interpretation of the large methane absorptions in the atmospheres of the major planets. Although only coefficients for the “random” or “Goody” band model are derived, the equivalence between these and the “regular” of “Elasser” coefficients is demonstrated. The effects of pressure on the absorption are surprisingly small, leading to large values of the pressure coefficient quite unlike any previous application of the band-model theory. They indicate a pseudo-continuum character of the methane spectrum throughout the visible and near infrared. A spectrum synthesis calculation using the derived coefficients shows the close fit to experimental data that can be realized.     

Non-linear flexural vibrations of orthotropic rectangular plates

This study is an analytical investigation of free flexural large amplitude vibrations of orthotropic rectangular plates with all-clamped and all-simply supported stress-free edges. The dynamic von Karman-type equations of the plate are used in the analysis. A solution satisfying the prescribed boundary conditions is expressed in the form of double series with coefficients being functions of time. The model equations are solved by expanding the time-dependent deflection coefficients into Fourier cosine series. As obtained by taking the first sixteen terms in the double series and the first two terms in the time series, numerical results are presented for non-linear frequencies of various modes of glass-epoxy, boron-epoxy and graphite-epoxy plates. The analysis shows that, for large values of the amplitude, the effect of coupling of vibrating modes on the non-linear frequency of the fundamental mode is significant for orthotropic plates, especially for high-modulus composite plates.     

The elastic dielectric

Macroscopic field equations, boundary conditions and equations of state are derived for the non-linear, macroscopic elastic and dielectric response of an insulator. A centrosymmetric polynomial representation of order four is introduced for the energy density; the equations of state for the electric field and stress tensor are then deduced as polynomials of degree three in the displacement gradients and electric displacement field. The results are applied to the special case of m3m material symmetry.

A finite, point-charge model of a centrosymmetric ionic crystal is introduced and used to determine 0°K microscopic expressions for the electric field and stress tensor equation of state coefficients introduced in the macroscopic analysis. The results are used to calculate the full set of second and third-order non-linear coefficients for NaI, based on a Born-Mayer potential and the 4·2°K elastic stiffness data of Claytor and Marshall.     

Selection rules forG 2 vectors in the atomicf shell

In various tabulations of such spectroscopic coefficients as the matrix elements of tensor operators or fractional parentage coefficients, it is found that many entries are unexpectedly zero. A survey is made of all cases that occur in the atomicf shell and that involve the 7-dimensional vector representation of the groupG ₂. Direct explanations are given in terms of the group structure of the electronic configurations that comprise the shell. The techniques used depend on a splitting of the state space into “spin-up” and “spin-down” parts, and, for other cases, the extensive use of the methods of second quantization. TheF terms of the atomicf shell are found to split into three classes. The separation that this classification provides for the twoF terms belonging to the irreducible representation (31) ofG ₂ coincides with Racah's separation. An improved separation of theH states of (31) is described.     

Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

In Parts I and II of this series of papers, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written on the modal basis, in the neighbourhood of each resonance, has been developed for beams and fully clamped rectangular plates.¹ Simple explicit formulae have been derived, which allowed, via the so-called first formulation, direct calculation of the basic function contributions to the first three non-linear mode shapes of clamped-clamped and clamped-simply supported beams, and the two first non-linear mode shapes of FCRP. Also, in Part I of this series of papers, this approach has been successively extended, in order to determine the amplitude-dependent deflection shapes associated with the non-linear steady state periodic forced response² of clamped-clamped beams, excited by a concentrated or a distributed harmonic force in the neighbourhood of the first resonance.This new approach has been applied in the present work to obtain the NLSSPFR formulation for FCRP, SSRP, and CCCSSRP, leading in each case to a non-linear system of coupled differential equations, which may be considered as a multi-dimensional form of the well-known Duffing equation. The single-mode assumption, and the harmonic balance method, have been used for both harmonic concentrated and distributed excitation forces, leading to one-dimensional non-linear frequency response functions of the plates considered. Comparisons have been made between the curves based on these functions, and the results available in the literature, showing a reasonable agreement, for finite but relatively small vibration amplitudes. A more accurate estimation of the FCRP non-linear frequency response functions has been obtained by the extension of the improved version of the semi-analytical model developed in Part I for the NLSSPFR of beams, to the case of FCRP, leading to explicit analytical expressions for the “multi-dimensional non-linear frequency response function”, depending on the forcing level, and the amplitude of the response induced in the range considered for the excitation frequency.     

General outline of the projector variation method

A method of “varying a projector” has been described recently. The term “projector variation” will be explained from a more general point of view, and certain extensions of the method are presented. Contrary to the Schrödinger variational principle, a pointwise determination of the wave function itself leads to a lowest error of second (not of first) order.     

Effects of collisions on linear and non-linear spectroscopic line shapes

A fundamental physical problem is the determination of atom-atom, atom-molecule and molecule-molecule differential and total scattering cross sections. In this work, a technique for studying atomic and molecular collisions using spectroscopic line shape analysis is discussed. Collisions occuring within an atomic or molecular sample influence the sample's absorptive or emissive properties. Consequently the line shapes associated with the linear or non-linear absorption of external fields by an atomic system reflect the collisional processes occuring in the gas. Explicit line shape expressions are derived characterizing linear or saturated absorption by two- or three-level “active” atoms which are undergoing collisions with perturber atoms. The line shapes may be broadened, shifted, narrowed, or distorted as a result of collisions which may be “phase-interrupting” or “velocity-changing” in nature. Systematic line shape studies can be used to obtain information on both the differential and total active atom-perturber scattering cross sections.