Conventional Linear Theory Research Articles

FORMULATION OF DEFORMATION BEHAVIOUR OF SOFTENED CELLULOID UNDER PROPORTIONAL LOADING BASED ON NON-LINEAR VISCOELASTIC THEORY

Recently, a non-linear viscoelastic theory was developed by extending the conventional linear viscoelastic theory. Especially, Ilyushin has proposed a principal quasi-linear viscoelastic theory in which multiple integral representation in the on-linear theory is rationally modified into a single integral one for convenience of evaluation. This theory is thus promising for analysis of the elastic-viscoplastic behaviour of high-polymer. In the present paper, for investigating the applicability of this theory instead of the constitutive equation hither-to use in the photo-rheologic stress analysis, a simple case of this theory was discussed comparing with results obtained from an experiment on a thin-walled tube of softened celluloid under torsion. As the result, the principal third-order viscoelasitc theory was found applicable to approximation of the experimental result with the same accuracy as with the constitutive equation hither-to used. It was also found that every stress-strain curve for combined loadings of torsion and tension could be transformed into that for the corresponding pure torsion, by modifying the stress space with consideration of invariants of stress tensor.

Giant polariton

The conventional linear theory of polaritons breaks down at high light intensities. We present the nonlinear equations for the exciton-photon coupling and a “polariton-like” solution which may contain a macroscopic number of excitons bound to photons.

Heterophase dislocations — An approach towards interpreting high temperature grain boundary behavior

When the conditions of full thermodynamic equilibrium are added to the conventional linear elastic theory of dislocations a hybrid concept of “heterophase” dislocations arises. As developed here, heterophase dislocation theory provides a self-consistent method for calculating the core configuration, energy, and stress-strain field which minimize the thermodynamic potential of a dislocated crystal. Simple grain boundaries in crystals near the melting point are treated as constrained arrays of heterophase misfit dislocations with a liquid-like core phase similar to, but not identical with the properties of ordinary liquid phase. Measurements of the properties and behavior of {011̄} tilt boundaries in bismuth crystals near the melting point show that the new theory accounts accurately for the energetic dependence on tilt misorientation in the range 0° to 6°, and then deviates from the observed dependence. The new theory also permits prediction of a structural transition in grain boundaries which is extremely sensitive to the density of misfit dislocations. This transition, predicted to occur in bismuth at a tilt misorientation of 15° was also confirmed by hot-stage electron microscopy.

Derivation of a linear theory for nonequilibrium and equilibrium flows

Conventional linear theory of nonequilibrium and equilibrium gas flows yields correct results only for very small deviations of the stream parameters from the unperturbed values. Moreover, if in linearization we take the coordinates in planar flow as independent variables, then the flow past concave and convex corners is described in exactly the same fashion. In this case the characteristic emanating from the corner is (depending on the type of corner) a compression or rarefaction shock. In the case of a break in the wall of an axisymmetric channel the shock intensity approaches infinity with approach to the centerline, which indicates a deficiency of this type of linear theory. In the following we use a modification which eliminates the deficiencies noted above. This involves conversion to new independent and dependent variables such that the coefficients of the exact equations being linearized become weakly varying functions of the unknown parameters, the linearized boundary conditions coincide with the exact conditions at all or part of the boundaries, and the rarefaction shocks become rarefaction wave bundles of finite width. The last condition is achieved as a result of the fact that, in accordance with the Lighthill method of deformable coordinates [1], we take as one of the independent variables a quantity which maintains a constant value on each characteristic of the bundle of characteristics emanating from the break point [for equilibrium flows the semicharacteristic (or characteristic) independent variables were used in deriving the linear theory, for example, in [2–4]]. The study was based on the example of two-dimensional stationary nonequilibrium flow of an inviscid and nonheatconducting gas. In this case we find that boththelinear equations at a finite distance from the walls and the boundary conditions for determining the potential and nonequilibrium parameters outside the rarefaction wave bundles coincide with the equations and the conditions of conventional linear theory [5], while the relations associating the values of the parameters on the closing characteristics of each bundle (outside the bundles the same value of the characteristic variable corresponds to these characteristics) at some distance from the axis or from some reflecting surface are identical to the conditions on the rarefaction shocks. This fact makes it possible to use several results of conventional linear theory.

Visco-elastic propagation of longitudinal waves in skeletal muscle

A method for obtaining the relaxation spectrum of isolated frog skeletal muscle from measurements of phase velocity and attenuation of longitudinal waves, was derived from conventional linear theory of visco-elastic wave propagation. The experimental method for measuring wave velocity and attenuation is described. An approximation method for obtaining the relaxation spectrum from wave velocity alone was developed from theoretical considerations. Experimental results from both the conventional and the approximation methods are presented and compared. Wave propagation measurements appeared to be of value in visco-elastic studies of skeletal muscle, especially when the use of other traditional methods is not practical.

Theory of a Turbulent Stationary State of a Plasma

We first establish a self-consistent scheme to determine the fluctuation spectrum for a class of turbulent plasmas in which a conventional linear theory predicts an exponential growth of the density-fluctuation excitations, or instability. This theoretical approach is based on the fundamental postulate that a proper inclusion of the correlation effects or the existence of the fluctuations in the stationary state should be able to remove the instability and lead to a description of the most stable state for the turbulent plasma. A dielectric response function $\ensuremath{\epsilon}(\mathbf{k},\ensuremath{\omega})$ for such a turbulent plasma is calculated within the validity of the hydrodynamic description. In this calculation, there are involved various polarization processes associated with the interaction between the external test charge (or the induced fluctuations) and the turbulent fluctuations existing in the plasma; the nature of those processes is clarified with the aid of diagrammatic considerations. The fluctuations of the internal electric field give rise to an additional mechanism for particle diffusion; the effective diffusion coefficient in the turbulent plasma is obtained by investigating the behavior of $\ensuremath{\epsilon}(\mathbf{k},\ensuremath{\omega})$ in the limit of long wavelengths and low frequencies. The effects of turbulence upon the properties of the ion acoustic wave are studied. Following the self-consistent scheme, an integral equation is derived for the fluctuation spectrum associated with the acoustic mode; it is then solved for values of the electron drift velocity ${\mathbf{V}}_{d}$ above and below the critical one, ${V}_{c}$. We thus find that the results indeed support our original postulate, and the dielectric response function remains stable for the entire range of the drift velocity. The over-all structure of the fluctuation spectrum is investigated. In terms of the small plasma parameter, $g\ensuremath{\equiv}\frac{1}{{n}_{0}\ensuremath{\lambda}{D}^{3}}$, the energy $\ensuremath{\epsilon}(\mathbf{k})$ in the acoustic mode with wave vector k is of the order of $g$ (i.e., around the thermal level) in the stable region; as the plasma enters the transition region, $\stackrel{^}{k}$. ${\mathbf{V}}_{d}\ensuremath{\simeq}{V}_{c}$, the order of $\ensuremath{\epsilon}(\mathbf{k})$ goes up to ${g}^{\frac{1}{2}}$; in the turbulent region, $\ensuremath{\epsilon}(\mathbf{k})$ contains a part of the order of ${g}^{0}$. It is also shown that a certain domain of the turbulence spectrum can be explained with the aid of a dimensional argument. The results of the calculations are compared with a fluctuation spectrum measured by a microwave scattering experiment.

Optimal Control of a Piecewise Linear System†

ABSTRACT A method of determining the control action of a piecewise linear system to minimise a quadratic cost function is presented. The piecewise linear system considered is described by a different set of linear differential equations for each of two regions of the state space ; the two regions are separated by a hyperplane. It is shown that, with certain restrictions, conventional linear control theory can be used, provided the time taken by an optimal trajectory to reach the separating hyperplane can be determined. A mothod for determining this time is given, thus completing the solution.

Experimental studies of sea-surface effects on low-angle radars

This paper presents results of measurements made during the years 1953?56 on the effects of sea-surface reflections on low-angle radars at Xband frequencies. The work was done in three parts and comprised a study of the polarisation properties of sea clutter as a function of sea state, wind direction and incidence angle, a study of the position and depth of interference minima close to the sea surface and a study of the reflecting properties of the sea over a wide range of incidence angles, in so far as these affect the properties of a possible image target. The sea-clutter measurements yielded a well marked polarisation pattern, but the effect of incidence angle did not conform to the conventional linear theory. The interference minima were located close to the positions predicted by the 4/3-earth theory when the aircraft flew at heights of 400ft or more. At lower altitudes, however, the range of the horizon and of the farthest minima considerably exceeded the theoretical values. The peak voltage-reflection coefficients over a range of angles between 1° and 40° relative to glancing incidence very seldom exceeded 0.3 for vertical polarisation and 0.6 for horizontal polarisation. Brewster-angle behaviour was clearly visible for all sea states, and the horizontally polarised reflection was always dominant. In all the results, features were observed for which no theoretical explanations are apparent.

Static and Sliding Friction in Feedback Systems

One of the most common nonlinearities encountered in servomechanisms design is the friction phenomenon in electromechanical systems. Conventional linear theory fails to predict its effect upon system performance. This paper extends familiar techniques to the treatment of friction nonlinearity in servosystems. Frequency-response methods are employed throughout and the theoretical results are verified by means of an analog computer. Sliding friction and static friction are represented by describing functions which form the critical factors in determining system stability. The analysis indicates that certain series equalizers designed from linear theory may fail to achieve effective compensation in the presence of sliding and static friction. On the other hand, a subsidiary loop may avoid the stability problem while still realizing an essentially equivalent loop gain function.