Dynamic forces in multiport mechanics: Direct and Lagrangian formulations

Holonomic systems can be represented by a bond graph in which inertial variables are related to generalized variables using a multiport displacement-modulated transformer structure. It is shown that the part of the generalized forces due to inertial elements can be found either by direct calculation or by the operations of Lagrange's equations. Although no gyrators are needed for the representation, gyrational coupling may exist. Alternative representations may involve gyrators explicitly and it is possible to construct kinetic co-energy functions for these gyrators for use in Lagrange's equations.     

On Constitutive Equations and Dimensional Analysis

A method of obtaining a system of partial differential equations from the invariance of a constitutive function under some restrictions of a continuous group of transformations is shown. Using the theory of partial differential equations, a set of independent solutions and dependence relationship is derived. Group theory technique is employed relative to dimensional analysis to show how the transformation from one unit system to another results in a relationship among different dimensional variables. An alternative proof of Buckingham's Pi-theorem is presented.     

On stability of a class of discrete systems

The problem of stability properties for the solutions of nonlinear difference equations is considered. The approach used is to study the behavior of the solutions of nonlinear difference equations with respect to solutions of a nonlinear difference equation. This is a more general setting than the comparison principle in which the comparison equation is a linear difference equation.The principal technique employed is an extension of Liapunov's direct method. A series of theorems is obtained yielding criteria for the behavior of solutions in terms of existence of the Liapunov-type function with appropriate properties.     

An integral-equation approach to problems of vibrating beams

In this paper integral equations are applied for the calculation of the normal modes of vibrating beams. Both exact and approximate methods of solving the integral equation are considered. The Green's function, or kernel, of the integral equation is constructed for both uniform and nonuniform beams. Solutions for the normal modes of a uniform cantilever are given. A nonuniform, naturally-twisted turbine blade is studied in detail and the first and second normal modes are calculated by the integral-equation method.     

On a Method of Solving Sensitive Boundary Value Problems

Nonlinear two-point boundary value problems have always been difficult to solve. The difficulty is compounded if the problem tends to be inherently unstable. This paper describes an algorithm for solving such sensitive boundary value problems. The procedure is based on a computational method for finding the general solution of systems of ordinary differential equations used in conjunction with the multi-point quasilinearization method of Miele. The method is demonstrated by solving Troesch's problem and a singular perturbation problem.     

Solution of integral equations using a set of block pulse functions

A set of the block pulse functions is applied to solve the Fredholm's and the Volterra's integral equations of the second kind. An algebraic equation in matrix form which is equivalent to the solution of the integral equation is developed. The approximate results are easily obtained by a few computations. An accurate solution canbe evaluated in a digital computer by solving the algebraic equation. Two examples are given.     

Solution of the Stochastic control problem in unbounded domains

Bellman's dynamic programming equation for the optimal index and control law for stochastic control problems is a parabolic or elliptic partial differential equation frequently defined in an unbounded domain. Existing methods of solution require bounded domain approximations, the application of singular perturbation techniques or Monte Carlo simulation procedures.In this paper, using the fact that Poisson impulse noise tends to a Gaussian process under certain limiting conditions, a method which achieves an arbitrarily good approximate solution to the stochastic control problem is given. The method uses the two iterative techniques of successive approximation and quasi-linearization and is inherently more efficient than existing methods of solution.     

Integraph solution of differential equations

In a previous article¹ a continuously recording integraph was described, by means of which differential equations, involving only one integration, could be solved. The present article describes a revision of this machine such that an equation involving two successive integrations, corresponding to practically any second-order total differential equation, with all terminal conditions included, can be solved. The need for a workable means of solving the differential equations involving empirical and discontinuous coefficients which occur repeatedly in electrical engineering and physics is recalled. In the machine described such solutions are effected by means of suitable interlinked integrating devices, the result being plotted continuously as a function of the independent variable. Tests and simple solutions show the over-all error to be approximately 1 or 2 per cent. The various sources of this error are discussed.     

On Maxwell's covariant equations in matter

This paper first presents a fully covariant formulation of Maxwell's equations in matter in the general relativistic framework. Although covariant, this formulation uses only essentially spatial four-vector fields and places in evidence the kinematical couplings (vorticity) and inertial effects (curvature of worldlines) which appear in Maxwell's equations for matter. Using a scaling of these equations enables one to construct covariant “quasi-electrostatic” and “quasi-magnetostatic” approximations. Theformer is used to establish the equations that govern piezoelectric gravitational-wave detectors. Furthermore, if a pyroelectric crystal is used in this type of device, then Gauss' law is shown to be modified and contains a contribution due to the incident gravitational perturbation.     

The probabilistic character of relevance

Gebhardt's[1] probabilistic model of relevance is examined and found not to represent adequately some characteristics of the relevance judgement process. An alternative model is proposed, which identifies two different types of “error” or probabilistic variation between relevance judgements. The two types arise from, first, the definition of the boundaries of the relevance classes, and secondly the actual assessment of an individual document on the underlying scale (which is assumed to be a continuum). The problems of quantifying the model, and of assessing its implications for retrieval testing, are discussed.     

A state space approach to elasticity

The two-dimensional, plane stress problem of linear elasticity is analyzed within a state space framework. The medium considered is homogeneous and isotropic. Vlasov's mixed formulation of elasticity is used throughout. The field equations are derived in closed form, thus avoiding Vlasov's intermediate infinite series solution. Finally, all the properties of the transfer matrix are shown to follow directly from embedding the problem into a state space setting.     

Pseudo bond graphs of circulating Fluids with application to solar Heating design

Two different bond-graph schemes for the modeling of circulating fluids are discussed and compared. The first uses a heuristic approach wherein the flow path is broken into segments. Each segment has separate hydraulic and thermal submodels which are linked with active bonds. The model is general enough to handle natural convective flows, heat storage, conduction, forced convective heat transfer and heat energy sources. The second method, applicable to pumped flows, uses a modal analysis approach to derive state equations for the modal temperatures, the time varying coefficients of the spatially varying mode shapes in a separation of variables solution for the temperature partial differential equation. Two examples are worked to illustrate the models. Numerical simulation results are presented using the first method and a simple analytic solution given to illustrate the second.     

Exact solutions for concentration dependent diffusion and the inverse complementary error function

A direct method of studying the qualitative behavior of concentration dependent diffusion is based on exact solution of problems with diffusion coefficients which are piecewise constant functions of concentration. Polynomial approximations are given for the inverse of the complementary error function. They facilitate the direct method and also Philip's inverse method of studying qualitative behavior.     

An efficient method for solving Emden–Fowler equations

In this paper, He's variational iteration method (VIM) is applied to solve the Emden–Fowler type equations in the second-order ordinary differential equations (ODEs). In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The Lagrange multipliers in the functionals can be identified optimally via variational theory. This technique provides a sequence of functions which convergence to the exact solutions of the Emden–Fowler equations. Comparison with the exact solutions and the solutions by the Adomian decomposition method (ADM) show efficiency of VIM in solving equations with singularity.     

Numerical solution of systems of mixed partial differential equations associated with transport phenomena with exchange

The so-called “percolation operations” which involve simultaneous heat and mass transport are defined in terms of a system of mixed partial differential equations which are nonlinear and include variable coefficients. Second-order interpolation followed by integration over an increment of the time or space variable is proposed as a method for arriving at finite difference equation equivalents of the original partial differential equations. A model change is also used to give physical significance to finite difference equivalents associated with derivatives in the space dimension. It is shown that second-order interpolation has the advantage over linear interpolation that it introduces a factor β which in a stability analysis based on cencentrations offers a range of choices that will insure stable calculation on a digital machine.     

The origin of dimensional analysis

A paper by Macagno (3) in this Journal is discussed. The origin of the concept of physical dimensions is traced back to ideas previously used in analytic geometry. Descartes' use of the word “dimension” in the study of physical magnitudes is shown to have properties completely different from Fourier's dimensions, being therefore unimportant to the evolution of dimensional analysis. It is also shown that the principle of homogeneity was used in the derivation of physical equations sixty years prior to the publication of Fourier's work, and that the latter was aware of this paper, that may be considered the earliest publication on dimensional analysis.     

Optimal stabilization for differential systems with delays - Malkin’s approach

The paper considers a process controlled by a system of delayed differential equations. Under certain assumptions, a control function is determined such that the zero solution of the system is asymptotically stable and, for an arbitrary solution, the integral quality criterion with infinite upper limit exists and attains its minimum value in a given sense. To solve this problem, Malkin’s approach to ordinary differential systems is extended to delayed functional differential equations, and Lyapunov’s second method is applied. The results are illustrated by examples, and applied to some classes of delayed linear differential equations.