Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1968"

A generalization of the Popov criterion

Abstract: A criterion for the stability of control systems which contain an arbitrary finite number of memoryless nonlinearities is considered. The criterion is such that the degree of stability may be specified, and such that for the case when the absolute stability of a control system having one memoryless nonlinearity is considered, it reduces to the original Popov criterion.

Dynamic stability of an axially moving band

Abstract: This paper determines approximate stability-instability region boundaries for two cases of parametric excitation. The first problem considers periodic, axial, tension variation of slender, axially moving materials such as band saws, belts, tapes, strings and chains; the second case investigates instability caused by periodic, in-plane, edge loading in axially moving materials. The governing equation of motion is reduced by means of a coordinate function expansion and Galerkin's method to a set of coupled Mathieu equations. The methods of Hsu and Bolotin are used to construct stability boundaries for the two cases. Results are compared with analog computer stability boundaries for a moving string; the string was spatially discretized by replacing spatial derivatives by equivalent difference expressions. Boundaries predicted by the two methods are close for moderate material axial velocities but separate as the axial velocity increases.

On the definition and application of the sensitivity function

Abstract: An exposition is presented of the application and scope of a new interpretation of the classical sensitivity function Sp T (s) that is used to evaluate the effectiveness of the over-all feedback system T(s). in reducing sensitivity to parameter deviations in the plant P(s). It is shown that this interpretation, due to Cruz and Perkins, when considered as the basic definition of the sensitivity function, is of a much broader application than the classical definition: it encompasses regulator as well as servo systems, scalar time-invariant plant-parameter deviations as well as time-varying multiparameter variations, single-.loop and multiloop systems, sensitivity to plant parameters as well as to disturbances. Discussion is confined to linear time-invariant systems, but the new concept of the sensitivity function is also applicable to time-varying and nonlinear systems.

Cosmic static at 144 meters wavelength

Abstract: A radio map of the southern sky shows a very bright background crossed by absorption clouds of ionized hydrogen in the plane of the Milky Way. The bright background may represent the energy extracted from light as it travels through intergalactic space. The shift in spectral lines of visible light may be explained without relative motion. Small radio sources and diffraction patterns are described. Ionospheric phenomena due to sporadic E, meteors and pulsations from the antarctic are discussed.

Nonlinear bending of beams with concentrated loads

Abstract: A simple numerical method is proposed for analyzing nonlinear bending of beams. The success of the application of this method to the nonlinear problems is demonstrated by two examples: (1) a cantilever beam with a concentrated load at the free end, and (2) a simply supported beam subjected to a nonsymmetrical load. The results are checked by other known solutions.

Analysis of control systems with complex nonlinearities and transportation lag

Abstract: When a control system contains a hysteretic nonlinearity, or a dead time element (transportation lag), or a distributed lag, the system may be described by a characteristic equation with complex coefficients which are functions of dependent variables. The characteristic equation is partitioned and root locus methods are used to analyze stability, existence and characteristics of limit cycles, damping characteristics of stable systems, etc. Control systems with frequency dependent, hysteretic, nonlinearities and with adjustable parameters are analyzed using parameter plane and parameter space methods.

Vibrations and stability of a moving band

Abstract: The problem of a thin strip moving with constant speed in a longitudinal direction is formulated. The effect of a conservative point load acting parallel to the plane of the strip and normal to one of the short sides of the cross section is included in the formulation and a study is made of the lateral and torsional motions of the strip. The equations governing the dynamic motion of the strip are derived by a variational method and it is shown that the presence of the point load couples the lateral and torsional motions both in the system differential equations and in the boundary conditions. The coupled equations have variable coefficients due to the terms involving the initial state of stress arising from the presence of the point load. An exact analytical solution not being feasible, a parametric study of the lowest modes is carried out using a collocation method of solution. The effects of various system parameters on the frequency are studied and the occurrence of a static buckling load is predicted. The results of the study are compared with a previous analysis where the coupling of lateral and torsional motion was neglected. It is shown that in certain cases, the coupling effect is very important and cannot be neglected.

Mechanisms of polarized and depolarized scattering from a rough dielectric surface

Abstract: A theory is developed to explain the nature of polarised and depolarized scattering of electromagnetic waves from a slightly rough dielectric surface. The method of small perturbation is used together with the Fourier transformation and results are carried up to and including the second-order. It is shown that first-order fields result from a single scattering process and second order fields result from a two-bounce multiple scattering process. It is also shown that regardless of the dielectric property of the surface, the incident wave number, the incident angle and the direction of the receiver combine to determine exactly the harmonic components of the surface that are responsible for scattering.

On observability of stochastic discrete-time dynamic systems☆

Abstract: Observability is defined for stochastic discrete-time dynamic systems as the existence of conditions on the system state vector estimates such that the error-covariance matrices have certain specified asymptotic properties. It is shown that stochastic observability can be characterized in terms of certain rank conditions which are generalizations of the well-known rank condition that characterizes the observability of deterministic linear time-invariant systems. As a by-product in the investigation of the asymptotic behavior of error covariance matrices, new upper and lower bounds are obtained on the error covariance matrices of the state vector estimates of linear discrete-time stochastic systems. Some information theoretic interpretations of the observability condition are also offered for the results obtained.

A generalized mathematical theory and experimental verification of proportional notches

Abstract: A theory and generalized synthesis procedure is advocated for the design of weir notches and orifice-notches having a base in any given shape, to a depth a, such that the discharge through it is proportional to any singular monotonically-increasing function of the depth of flow measured above a certain datum. The problem is reduced to finding an exact solution of a Volterra integral equation in Abel form. The maximization of the depth of the datum below the crest of the notch is investigated. Proof is given that for a weir notch made out of one continuous curve, and for a flow proportional to the mth power of the head, it is impossible to bring the datum lower than (2m − 1)a below the crest of the notch. A new concept of an orifice-notch, having discontinuity in the curve and a division of flow into two distinct portions, is presented. The division of flow is shown to have a beneficial effect in reducing the datum below (2m − 1)a from the crest of the weir and still maintaining the proportionality of the flow. Experimental proof with one such orifice-notch is found to have a constant coefficient of discharge of 0.625. The importance of this analysis in the design of grit chambers is emphasized.

Transient heat transfer in human skin

Abstract: Experiments are discussed which assess the influence of sudden, localized changes of thermal load on the conductance of human skin; the conductance is expected to change due to the thermal regulatory role of skin. The changes are produced by placing local areas of skin in contact with passive probes which are pre-heated or pre-cooled. It is found that the change in conductance is effectively independent of the surface temperature (and of the surface heat flux) to which the skin is exposed at the beginning of each test, for elapsed times exceeding one minute.

Structural analysis by matrix decomposition

Abstract: A physical interpretation is derived for the Choleski decomposition method as applied to structural analysis. Both the stiffness (displacement) and flexibility (force) methods of structural analysis are treated. The interpretation is shown to be valuable in providing physically meaningful upper and lower bounds on the elements of the decomposed diagonal matrix. These bounds are useful in error analysis of structural computations on a digital computer.

Flexible bars subjected to arbitrary discrete loads and boundary conditions

Abstract: A method of analysis for the large deflections of piecewise prismatic elastic bars subjected to an arbitrary number of discrete loads is presented. By combining the general solution of the nonlinear differential equation of bending and the equilibrium equations for each of the intervals between the load points and points of discontinuity in the flexural rigidity, along with the given boundary conditions, the problem is formulated as a system of simultaneous nonlinear equations which is solved by a modified Newton-Raphson iteration procedure. Four examples are given to illustrate the application of the method for the treatment of beam-columns with various loading and boundary conditions.

Properties of some Hugoniot curves associated with shock instability

Abstract: Combining the Hugoniot equation e e 0  1 2 (p + p 0 )(v 0  v) with the (e-p-v) equation of slate gives the Hugoniot curve ph(p0, v0, v) centered at an initial condition (p0, v0) in the (p-v) plane. Since the Hugoniot equation satisfies the conservation laws of mass, momentum, and energy, only those Hugoniot curves that are consistent with the second law of thermodynamics define the locus of states connected to (p0, v0) by a single compression shock. Hugoniot curves that lie in a domain of the (p-v) plane where ( ∂ 2 p ∂v 2 ) 8 > 0 are consistent with the second law of thermodynamics; those that lie in a domain where ( ∂ 2 p ∂v 2 ) 8 > 0 , as well as in a domain where ( ∂ 2 p ∂v 2 ) 8 , are inconsistent with the second law and are associated with shock instability. The first law of thermodynamics gives properties of some Hugoniot curves associated with shock instability. The second law of thermodynamics, together with conditions necessary for hydrodynamic stability, gives the locus of states connected to a given initial condition in the domain of the (p-v) plane where a single compression shock is thermodynamically unstable.

Development and evaluation of transfer functions for nucleate, transition and film boiling

Abstract: This study is concerned with the dynamic behavior of a heat transfer process which is described in terms of small variations in heating element surface temperature and input power in the vicinity of any operating point on the boiling curve. Responses are very nearly linear during such operation, thus transfer function concepts can be applied. Transfer functions relating small increments of heating element surface temperature and power input are developed analytically for a cylindrical heating element immersed in liquid, and the extension to ribbon type heating elements is described. Transfer function parameters are evaluated and the manner of variation of these parameters as the boiling curve is traversed, is demonstrated.

Minimizing system sensitivity through feedback

Abstract: A function space representation is used to examine the problem of reducing system sensitivity by means of feedback. It is shown that this question leads naturally to the problem of minimizing an abstract sensitivity index. In the course of the study earlier results of Cruz ( 3 ) and Zames ( 15 ) for stationary differential systems are extended to cover large classes of nonstationary discrete, distributive and composite systems.

Bending and twisting of thick plates with a circular hole

Abstract: A closed-form solution is obtained for the elastic bending of an infinite plate with a circular hole, the solution is based on a three-dimensional thick plate theory developed earlier by Lee and Donnell (7). Numerical results of the stress concentration factor obtained here are compared to those based on Reissner and classical thin plate theories. The three-dimensional series solution given by Alblas is discussed. Twisting load is handled as usual by superposition of bending loads. A simple formula relating the stress concentration factors of bending and twisting is reached, and numerical values of twisting stress concentration factor are included.

An exact solution to the elastic deformation of a finite length hollow cylinder

Abstract: An exact elastic analysis is presented for the deformation of a finite length hollow cylinder subjected to thermal and mechanical loads. The solution presented satisfies the surface and end boundary conditions identically. A comparison of the present solution with the classical solution shows that the latter is exact for a certain class of materials.

Continuum model study of preview effects in actively suspended long trains

Abstract: Previous work has shown that there can be a significant improvement in the riding qualities of vehicles traveling at high speed over stochastically described rough roadways if information about the roadway profile ahead of the vehicle is properly utilized. In this study, this “preview” of the roadway ahead of a vehicle is obtained from the dynamic variables of preceding cars in a long train. The analysis is simplified by considering a continuous version of a long train and studying partial differential equations rather than a large set of simultaneous ordinary differential equations. The results have application to the design of active suspensions for vertical motion and may be readily extended to active roll control systems which could anticipate changes in track curvature for guided vehicles.

Fourier analysis on wiener measure space

Abstract: The problem of representation of nonlinear systems on abstract spaces by a complete set of orthogonal functions defined on the same space was partly solved by Wiener, et al. (1–4) for nonlinear time invariant systems on the Wiener measure space (ΣI, BI, μ). This paper gives a simplified exposition of certain well-established results of Wiener and others (1, 6, 7, 8) in terms of non-rigorous concepts such as delta functions and white noise process in order to make the theory accessible to those knowing engineering mathematics. Proofs of Bessel's inequality and the Riesz-Fischer theorem which correspond directly to the modified Wiener's Orthogonal Set (9) are believed to be a contribution of this paper.

Shielding of transmission lines against lightning

Abstract: A simple geometric procedure to evaluate the shielding effect of earth wires on power transmission lines is discussed. The procedure is applicable not only to vertically moving lightning strokes but is valid also for strokes moving in any other direction. Correction factors are introduced in order to take into account the position and the cross-section of the earth wire relative to the phase conductors. The lightning-performances of three existing transmission lines (230 kv, 287 kv, and 345 kv) are compared with those predicted by the proposed method.

Some new re-lines with special functions for their solution

Abstract: An elementary transformation is used to reduce to a simple form, the Telegrapher's equation of a general tapered re-line. With suitable interrelationships between the distribution of r and c, the transformed equation is then solved in terms of the special functions such as Hermite, Bessel, confluent hypergeometric and Whittaker functions, resulting in a number of new classes of distributions. The effect of leakage is also discussed.

Pressurization of two layered incompressible orthotropic tubes

Abstract: The effect of internal pressure an a closed tube of incompressible orthotropic material is analyzed. The cross section of the tube consists of two regions each having a different set of material properties. An exact solution for the stresses, strains and radial displacement is obtained based on the theory of orthotropic elasticity. The exact solution is simplified for thin walled tubes. The presence of a second layer is shown to reduce the stress in the original layer by more than the ratio of wall thickness. In general, the stresses in each layer are unequal. A relation is presented which assures the existence of equal circumferential stress in each layer

Stress waves produced by impact on the surface of a plastic medium

Abstract: This paper discusses certain observations of the effect that local plastic yielding has on the radial surface strain waves produced by the impact of a hard sphere on the surface of a steel block. At impact velocities slightly greater than that required first to initiate local plastic yielding, the effect is almost imperceptible close to the area of impact, but becomes observable at distant points. At higher velocities of impact, however, the experimental results clearly show the presence of plastic yielding as a sudden change in the slope of the strain waves for all distances of travel. In the development of the theory the surface displacements of an elastic half-space are written as integrals over the first derivative of an arbitrary vertical loading f(t) applied at a point on the free surface. It is shown that at large distances r away from the point of impact the amplitude of the radial surface strain generated by f(t) is inversely proportional to r 1 2 . However, the radial surface strains excited by finite jumps of f(0) and f'(0) are shown to decay as r−2 and r−1, respectively, for large values of r. On the basis of the theory developed here, the slope of the applied forcing function is shown to vary rapidly when local plastic yielding occurs.