Approximate Linear Equation Research Articles

NON-LINEAR VIBRATION ANALYSIS AND SUBHARMONIC WHIRL FREQUENCIES OF THE JEFFCOTT ROTOR MODEL

In this analysis, a Jeffcott rotor model is used which is a thin disk located on a flexible shaft which is simply supported at the ends. The non-linear dynamic equations of the rotor are obtained. A perturbation technique is used to obtain approximate linear equations for the non-linear equations. The non-linear equations and approximate linear equations are solved numerically and the solutions compared. The approximate linear equations correctly predict the non-linear vibration. It has been experimentally observed in many researches that in addition to the synchronous whirl, there exist subharmonic vibrations which may cause instability. This is generally attributed to the dry friction, non-linear or asymmetric stiffness, rubs, fluid-film bearing clearances. This study shows that there exist two subharmonic transient vibrations caused by the non-linearity of the system itself. The two subharmonic frequencies are equal to (ω+ωn) and (ω−ωn) and also the supersynchronous component of the vibration becomes unstable when the speed ratio ω/ωnis ⩾2.

The Complete Integrability of the Linear Approximation Equations for Completely Integrable Quasilinear Total Differential Equations

The Complete Integrability of the Linear Approximation Equations for Completely Integrable Quasilinear Total Differential Equations

A simple and efficient method for diagnosing equipment faults based on quantitative physical knowledge

This paper proposes a method for identifying and locating the origin of a fault of a piece of equipment using approximate linear quantitative equations of variable increments which are derived from equations representing the steady state of the equipment and measured values of the variables. The proposed method differs fundamentally from strict numerical simulation methods and qualitative methods. Although the solution which is obtained by the proposed method is approximate, it is devised so that errors which are contained in them can be reduced. The proposed method, because only linear equations are dealt with, has the advantage of being easy to implement and process and of taking much less computation time.

On solutions to Laplace’s tidal equations in linear approximation

On solutions to Laplace’s tidal equations in linear approximation

Elastic biaxial bending and torsion of thin-walled members

This paper presents a detailed treatment of the non-linear elastic biaxial bending and torsion of thin-walled open section members. The treatment is valid for uniform members of linear elastic material, and is limited to small strains and rotations, and moderate deflections. Shear straining of the mid-surface of the member wall is neglected, and it is assumed that the member does not distort or buckle locally. The effects of initial deformations, loads, stresses, and strains are incorporated. The treatment is based on non-linear strain-displacement relationships, and these are used to derive the non-linear equilibrium and tangent stiffness equations in forms which are suitable for computer solution by the finite element method. Approximate linear and non-linear differential equilibrium equations are derived, as are the differential equilibrium equations and the energy equation for neutral equilibrium at bifurcation buckling, and these are then related to the classical equations developed by Timoshenko, Vlasov, and others.

Asymptotic analysis of the linear vibrations of a polymeric crystal

The authors have obtained analytically for the skeletal vibrations of the orthorhombic crystal of PE the asymptotics of the dispersion curves and approximate linear equations corresponding to small, large and medium phase shifts along the chain. The values of the energy gaps in the spectrum are calculated. The analytical solutions are compared with the results of a computer count and experimental data.

Roof deflections and sag in jointed, horizontally bedded strata ? A numerical study

A nonlinear finite element model with discrete joint modelling capability is developed. A semi-empirical equation introduced for determining the maximum deflections of clamped and vertically jointed roof beam is verified using a nonlinear finite element model. The mechanism of stability is also investigated numerically and concluded that the formation of Voussoir arch (Parobolic compressive zone) plays a major role in the overall stability of beam roofs over underground openings. Approximate linear equations are obtained for three joint spacings of 800, 400 and 200 mm, relating the maximum deflection of the beam and joint width for a given joint spacing.

Torsional vibration of shafts coupled by mechanisms

The paper is concerned with torsional vibration of a system composed of two parallel shafts coupled together by a number of mechanisms. The analysis has been done by using both non-linear and approximate linear equations. It is shown that there are three predominant sources of vibrations: (i) the tendency to different speed fluctuations of each mechanism; (ii) difierent mode configurations of the shafts; (iii) the fact that the inertia and the stiffness of the system are not varying proportionally to one another. The instability regions, the vibration amplitude and the range of fluctuation of the driving torque for a double crank-and-rocker mechanism are illustrated graphically.

The regularization of sets of linear inequalities

It is shown that the discrepancy principle proposed in /1–3/ to solve approximate linear operator equation can be used to construct a stable approximation to a normal solution of a set of linear inequalities. A single-parameter regularization algorithm for linear programming problem is constructed as an application.

On the Excitation of Oscillations of the Sun (Numerical Models)

AbstractNumerical solutions of the general time-dependent gas-dynamical equations in linear adiabatic approximation are given for initial conditions imitating: (a) a central perturbation, (b) a boundary perturbation (in the convective envelope), and (c) a ‘shrinking’ of the Sun as a whole. For a variety of models of the Sun it is found that at the surface the radial component vr of velocity is much greater than the tangential component vt, and that the period T of stationary oscillations does not exceed 131m. The appearance at the surface of a g mode with period 160m is found to be improbable.With the initial conditions adopted, a propagating wave is produced which is reflected successively from the centre to the periphery and back, producing 5-min oscillations at the surface of the Sun. Expansion of this wave into separate modes leads to a power spectrum qualitatively similar to that observed.

Calculation of Substitutional Impurity Distribution by Dissociative Mechanism

The basic equations for the dissociative mechanism of diffusion are a set of non-linear differential equations. Sturge obtained an approximate linear differential equation for the substitutional impurity distribution and solved it approximately (Proc. Phys. Soc. 73 (1959) 297). In this paper, Sturge's equation is solved numerically and the result is compared with his approximate solution. It is found that the approximate solution agrees fairly well with the numerical solution.

Angular Momentum Coordinates and Their Use in Zonal, Geostrophic Motion on a Hemisphere

The equations of motion for balanced, axisymmetric motion on a hemisphere are transformed to a somewhat simpler form employing angular momentum as a new, meridional coordinate. The resulting equations may be regarded as a relative of the semi-geostrophic equations and have similar properties such as the stretching of zones of high vorticity. Wind and temperature fields may be obtained by solving an approximate linear equation given the Ertel potential given vorticity field and suitable boundary conditions.

Dynamics of stationary ultra-long waves in middle latitudes

The dynamics of stationary ultra-long waves in middle latitudes are studied using a set of approximate linear equations for small perturbations about a zonal flow. the equations are derived by an expansion procedure based on the assumptions of planetary scales of variation and small mean superrotation. the zero-order equations reduce to a vertical structure equation containing no meridional derivatives. Analytical solutions are obtained for forced stationary waves in a model atmosphere consisting of three continuous layers, representing the troposphere, the lower stratosphere and the upper stratosphere. The waves are generated by thermal and topographic forcing in the model troposphere and propagate wave energy at zonal wavenumbers 1 and 2 into the upper stratosphere, where they are damped by Newtonian cooling. General relationships are found involving the zero-order fluxes of sensible heat and wave energy. These greatly clarify the dynamics of the waves, particularly in the lower stratospheric layer which links the regions of generation and dissipation. Taking realistic values of the parameters for northern middle latitudes in winter, the solutions for wavenumbers 1 and 2 show good agreement with observation. an examination of the sensitivity of the solution to changes in various model parameters shows that the meridional flux of heat by ultra-long waves in the troposphere is extremely sensitive to changes in the stratospheric wind profile and static stability. Since these waves are known observationally to account for a large fraction of the total poleward heat transport in northern middle latitudes in winter, a dynamical mechanism is suggested whereby variations in solar ultraviolet radiation or changes in the ozone content of the upper atmosphere could lead to climate change.

Dynamics of stationary ultra‐long waves in middle latitudes

AbstractThe dynamics of stationary ultra‐long waves in middle latitudes are studied using a set of approximate linear equations for small perturbations about a zonal flow. the equations are derived by an expansion procedure based on the assumptions of planetary scales of variation and small mean superrotation. the zero‐order equations reduce to a vertical structure equation containing no meridional derivatives. Analytical solutions are obtained for forced stationary waves in a model atmosphere consisting of three continuous layers, representing the troposphere, the lower stratosphere and the upper stratosphere.The waves are generated by thermal and topographic forcing in the model troposphere and propagate wave energy at zonal wavenumbers 1 and 2 into the upper stratosphere, where they are damped by Newtonian cooling. General relationships are found involving the zero‐order fluxes of sensible heat and wave energy. These greatly clarify the dynamics of the waves, particularly in the lower stratospheric layer which links the regions of generation and dissipation.Taking realistic values of the parameters for northern middle latitudes in winter, the solutions for wavenumbers 1 and 2 show good agreement with observation. an examination of the sensitivity of the solution to changes in various model parameters shows that the meridional flux of heat by ultra‐long waves in the troposphere is extremely sensitive to changes in the stratospheric wind profile and static stability. Since these waves are known observationally to account for a large fraction of the total poleward heat transport in northern middle latitudes in winter, a dynamical mechanism is suggested whereby variations in solar ultraviolet radiation or changes in the ozone content of the upper atmosphere could lead to climate change.

Method of successive approximations for linear equations of the first kind in Hilbert space

Method of successive approximations for linear equations of the first kind in Hilbert space

The method of successive approximations for linear equations in hilbert space

A method of successive approximations is constructed for linear equations of the form ƒ  (E - S) ϑ with an arbitrary non-increasing operator S acting in the Hilbert space H.

A simpler form of the general Reynolds equation : P. F. Fowles, JOLT, 92, Ser. F(4)(1970) 661–662; 3 refs.

In this analysis, a Jeffcott rotor model is used which is a thin disk located on a flexible shaft which is simply supported at the ends. The non-linear dynamic equations of the rotor are obtained. A perturbation technique is used to obtain approximate linear equations for the non-linear equations. The non-linear equations and approximate linear equations are solved numerically and the solutions compared. The approximate linear equations correctly predict the non-linear vibration. It has been experimentally observed in many researches that in addition to the synchronous whirl, there exist subharmonic vibrations which may cause instability. This is generally attributed to the dry friction, non-linear or asymmetric stiffness, rubs, fluid-film bearing clearances. This study shows that there exist two subharmonic transient vibrations caused by the non-linearity of the system itself. The two subharmonic frequencies are equal to (ω+ωn) and (ω−ωn) and also the supersynchronous component of the vibration becomes unstable when the speed ratio ω/ωnis ⩾2.

Methods of successive approximation for linear equations in Banach spaces

Methods of successive approximation for linear equations in Banach spaces

Das makroskopische Gravitationsfeld in der einheitlichen Quantenfeldtheorie

AbstractA theory of the gravitation field is proposed in connection with a new spinorial formulation of EINSTEIN'S equivalence principle and HEISENBERG'S unified quantum theory of fields. In the gravitation theory the gravitational field is given by the coefficients hAi (xl) of the matrix of the anholonomic non‐Lorentzian transformations from the Lorentzian metric ηAB to the physical metric gik (xl) of the Riemannian space‐time V4. The field equations and the commutation rules for the hAi are working in the Lorentzian space. The metric of the Riemannian V4 is given by the average gik = 〈|hAi, hBk|〉ηAB. For all matter fields (and for HEISENBERG's “Urfeld”) the field equations are equations in this Riemannian V4. The interaction between the gravitational field hAi and the matter‐tensor is given by a potential‐like coupling‐term.We prove that this new theory of gravitation gives for makroscopic matter the same linear approximation equations like general relativity and that a perihel‐motion δφ ≈ 76 δφ EINSTEIN results in the second approximation. But, we are finding essential differences to general relativity for the theory of gravitation radiation and for the theory of interaction of matter with strong gravitation fields.

A mathematical analysis of digitalis kinetics in patients with normal and reduced renal function

The kinetics of digitoxin and digoxin in patients with normal and reduced renal function have been examined by chemical determinations of urinary glycoside excretion and of endogenous creatinine clearance. Urinary digitoxin and digoxin excretion are shown to be linear functions of creatinine clearance. These findings permit reasonably accurate prediction of overall apparent biological half-time of digitoxin or digoxin in patients as a function of creatinine clearance. These predictions have been tested and found to be reasonable. Simple linear approximate equations for bedside use are presented for various ranges of apparent half-time. In addition, logical extensions of these kinetic relationships permit description of a general dosage formula for smooth replacement of one cardiac glycoside with another that has a different apparent half-time.