Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1970"

Stability of circular plates subjected to moving loads

Abstract: The Green's function for a centrally clamped, peripherally free plate is formulated as a Fourier eigenfunction series. The plate response is determined for two circumnavigating, peripheral-moving load excitations: (1) a harmonic load at constant angular velocity and (2) a load whose speed is the sum of a constant and a harmonic component. Circular plates containing radially symmetric membrane stresses and thickness are shown to possess exact stability criteria of a form identical to those of the classical plate even though the Green's function in these plates is unknown. Experiments were performed that support the theoretical developments.

Moving cracks in a finite strip under tearing action

Abstract: This paper is concerned with the anti-plane stress distribution around a semi-infinite crack traveling with constant velocity in an infinitely long strip of finite width. The problem is reduced to the solution of the Riemann-Hilbert problem by application of the Schwarz-Christoffel transformation and the theory of complex functions. Closed-form solutions are obtained for two cases of practical interest: (1) the boundaries of the strip are clamped and displaced in equal and opposite directions causing a tearing motion along the leading edge of the crack and (2) the crack is sheared longitudinally by a pair of concentrated forces moving with the crack while the strip boundaries are free of tractions. In both cases, the effect of strip width on the dynamic stresses is examined. A dynamic crack extension force is defined by a line integral the value of which is independent of the path of integration and requires only a knowledge of the local stress and displacement fields. The information gained in this study is useful in that it can assist interpretation of results from dynamic experiments on moving cracks.

On state estimation for distributed parameter systems

Abstract: Sequential algorithms for prediction, filtering and smoothing are developed for a class of linear distributed-parameter systems. The class of systems concerned is that involving noisy measurement data which are obtained from “averaging” and “scanner” -type sensors. The basic tools of the development are the least-squares estimation viewpoint, the calculus of variations and the sweep method for two-point boundary-value problems. An example involving the heat equation is presented to illustrate the results.

Variational solution of axisymmetric fluid flow in tubes with surface solidification

Abstract: The problem of axisymmetric heat conduction with internal surface solidification in the inlet regions of tubes is discussed. An approximate analytical solution is presented to this nonlinear, two-dimensional free-boundary problem. The analysis employs a variational technique which extends the Lagrangian formalism to treat the internal flow, two-dimensional moving-interface problems. The solution is expressed in terms of the short-time, asymptotic and steady-state components. Two forms of the variational solution are presented. One has limited validity in the entrance region of the tube, and the other, while less general, is more accurate. The results are compared with a finite-difference numerical solution. Agreement is of the order of 15 per cent.

Input—output properties of multiple-input, multiple-output discrete systems: part I☆

Abstract: Nonlinear time varying discrete feedback systems input-output properties, deriving stability criteria by generalized small gain and passivity theorems

Dynamic response of single cables with initial sag

Abstract: Governing equations for the plane motion of a single cable oscillating between two fixed end supports are simplified to study the specific case of small oscillations about an initially deformed state. It is shown that for small but finite initial sag, the governing equation can be cast in a form where the influence of initial sag depends on a single parameter. A closed-form solution for mode shapes and frequencies of an undamped cable can be obtained for arbitrary values of this cable parameter, and it is shown that significant changes in the lowest natural frequency are possible within the practical ranges of the cable parameter.

The physical basis of adhesion

Abstract: A general review is given of the main factors that determine the strength of adhesive joints. No type of atomic or molecular bond alone is strong enough to give a peel strength higher than about an ounce per inch width. Practically useful strengths are obtained by extending the range of action of the adhesive force by viscoelastic, viscous or plastic deformation to orders of magnitude higher than those of interatomic forces, or by a self-arresting mechanism of cracks in the case of brittle or nearly brittle adhesives. The crack may be arrested by internal stresses in the adhesive layer that guide the crack out of the layer, or by a process that reduces the stress concentration at the crack tip as the crack runs. Wetting of the adherend by the adhesive has less basic importance than is usually ascribed to it; surface energies, in particular, are not directly connected with the strength of the adhesive joint, as is seen from the fact that even with the highest possible specific energies of separation of the adhesive from the adherend the peel strength may be very low if neither viscoelasticity or plasticity, nor an effective crack-deflecting or crack-stopping mechanism is present. In an Appendix, a simple energetic derivation of Young's equation for the contact angle is given.

The role of ductility, tensile strength and fracture toughness in fatigue

Abstract: Fatigue process analysis through crack initiation and propagation and final fracture, considering ductility, tensile strength and fracture toughness roles

Final-value control of nonlinear composite distributed- and lumped-parameter systems

Abstract: The optimal final-value control problem for a general class of fully nonlinear composite distributed- and lumped-parameter systems is treated. The solution is derived with the aid of functional analysis techniques, together with an application of Green's identity, and is formulated in the form of a minimum principle which is a generalization of that presented by Katz. An iterative computational algorithm for implementing the results is developed and an example is given which illustrates the theory.

Fracture strength of relatively brittle structures and materials

Abstract: In the case of relatively brittle structural materials, linear fracture mechanics provides an analysis basis for testing and control of fracture strength. Standardized methods of crack toughness testing are available only for metals which can be pre-cracked by fatigue. Hence, further work on this topic is needed for glassy and polymeric solids. Proper use of surface residual compressive stress is a primary method of fracture control for such materials, and environment-assisted stable crack extension is often a significant factor. In the case of layered or parallel-fiber stiffened composites, the single crack fracture mechanics viewpoint usually applies to individual separations, but the aggregate effect of the expected distribution of cracks must be taken into account. Often the proof test is the main safeguard against strength-type failures in service, and the efficiency with which proof testing is employed is important.

On the application of perturbation to free-boundary problems in radial systems

Abstract: A first-order perturbation solution is obtained to a circular free-boundary problem associated with phase transformation. The solution describes the growth and decay of a solid layer during thermal interaction with a non-melting cylinder at sub-freezing temperature. The cylinder, which is initially at T0, is suddenly immersed in an infinite bath of liquid at temperature Ti with a freezing temperature Tf. The perturbation solution is compared with experimental data.

Heat transfer in composite media subject to distributed sources, and time-dependent discrete sources and surroundings

Abstract: The equation for the temperature distribution in each of k sections of a composite is solved. The composite consists of k discrete plates, cylinders of spheres, each of different material solidly joined at their k-1 interfaces. The composite media have distributed sources, and also have k-1 discrete sources at the interfaces. The composite media additionally have an arbitrary initial temperature distribution, and are exchanging heat at their mutual external boundaries with two different, arbitrary time-dependent surroundings through two different arbitrary constant film coefficients. The solution is obtained by means of a substitution that reduces the problem to that of solving a partial differential equation with homogeneous external boundary conditions. The solution is finally developed by means of a Vodicka type of orthogonality relationship.

On the prevention of delayed time failures of aerospace pressure vessels

Abstract: It is shown that proof testing a vessel can provide assurance of subsequent service life; however, the test can cause subcritical flaw growth. This growth must be minimized through careful attention to the test procedures. Several case histories are described, along with experimental static fracture toughness and sustained stress flaw growth data. These cases include such materials as aluminum, steel and titanium alloys, in various media, such as liquid propellants and test fluids. Times to failure and rate of growth of flaws are discussed in the context of linear elastic crack tip stress intensity factors. The influence of hold time at maximum load on the cyclic growth of flaws is also shown. Fracture mechanics analyses, together with static and sustained load fracture specimen tests, and the conventional proof pressure test of the vessel can be, and have been, used successfully in the prediction of minimum pressure vessel lives and modes of failure.

Dimensional properties of optimal and sub-optimal nonlinear control systems

Abstract: The optimal control signal for saturating or relay control systems is usually a complicated function of several state variables. A sub-optimal control method is to use instead a simpler signal which is a linear combination of functions each of which has only one state variable as argument. Persson has investigated this type of sub-optimal control for plants consisting of a set of pure integrators. In the present paper the theoretical basis of Persson's controller is developed by invoking dimensional considerations. It is shown that the optimal system has certain types of invariance under changes of scale. If a similar property for the sub-optimal system is adopted, the controller design problem can be reduced to the determination of a few parameters.

Atomic processes in the polar cap

Abstract: Polar cap atomic processes stimulated by photons, electrons and protons, considering particle morphology

Construction of Liapunov functions

Abstract: Liapunov functions construction through conversion of differential equation with polynomial nonlinearities into auxiliary exact differential equation using algorithm

An application of functional analysis to time optimal control of linear discrete systems with output constraints

Abstract: The problem of time optimal control of linear discrete systems with output constraints is formulated as an L-problem in the theory of moments. The solution to this latter problem is well known and is obtained from a finite-dimensional minimization. This procedure allows us to handle constraints not easily treated by other methods.

On the stability of Hill's equation☆

Abstract: By means of a modification of Malkin's method, the stability of the solutions of Hill's equation is studied. The procedure utilized is such that secular terms never appear in the development. From a first approximation, a simple expression is obtained for the boundaries of the stable regions in parameter space. A second approximation is also presented and applications are made to Mathieu's equation.

Formulas for the number of trees in certain incomplete graphs

Abstract: A concise summary is given of the standardized incomplete graphs denoted as the r, p, m and s series. Bercovici’s recent general formula for the number of trees in the m series is considered and the corresponding general formula for the s series is given.

On the drawing process of thin-walled tubes of Anisotropic material

Abstract: The influence of initial anisotropy on the reduction of thin-walled tubes is analysed. In the analysis, an upper-bound load and a lower-bound load are considered. The die is considered rigid, and a linearized yield condition with associated flow rules is used. It is shown that certain materials are less favorable than others in the drawing process. A set of equations is presented, which may be applied to find thickness variation, given material constants and die configuration.

Optimal control of nonlinear discrete time systems by mathematical programming

Abstract: A mathematical programming approach to the computational realization of optimal control problems for nonlinear discrete time systems is discussed. The method proposed is applicable to uniformly or nonuniformly sampled systems. The sampling intervals may be known or unknown a priori. The method includes any additional state-space, control and time interval constraints. The constraints may generally be nonlinear, of equality or inequality type. Several examples of implementing this method are presented. In addition to deterministic discrete time systems, stochastic systems are considered as well. An algorithm which implements mathematical programming solutions on a sequential basis is proposed.

Differential equation associated with electromagnetic waves in an inhomogeneous medium where ∇ε(r)·E vanishes

Abstract: Series solutions are obtained for a second-order ordinary differential equation associated with propagation of electromagnetic energy in an inhomogeneous plasma which varies in one direction only (say z ) for the case where E z = 0. The electron density and collision frequency distributions are represented by two different polynomials. When the collision frequency is a constant and the electron density, N( z ) varies as a first- or second-degree polynomial in z , the resulting differential equations can be represented by known functions, such as Airy functions or confluent hypergeometric functions, respectively. It is shown that the general series solution of the differential equation for which the electron density is also an arbitrary polynomial can be reduced to the aforementioned transcendental functions in the respective special cases.

Application of the Fokker-Planck-Kolmogorov equation to the analysis of differential pulse code modulation systems

Abstract: A representation of a Differential Pulse Code Modulation (DPCM) system with a uniform quantizer is given such that the “slope overload” and “granular” effects are separated. This representation includes a continuous feedback loop describing the “overload” effect followed by a parallel connection of appropriate samplers and quantizers, describing the “granular” effect. The granular noise is known to have approximately a flat spectrum over the signal band and a uniform probability density function. The spectrum of the slope overload noise as well as its probability density function is not known. Using the above representation a study of the probability density function of the slope overload noise is presented for an input which is a vector Gauss-Markov process. This study uses methods known in control theory. Explicit results are presented for certain input processes with simple spectra. Some applications of continuous Markov processes to this problem are discussed.

Construction of redundant networks with minimum distance

Abstract: A method for constructing redundant networks and simultaneously minimizing total distance is presented. The method is well suited for digital computation. Its application to message switching systems is briefly discussed.