An optimum technique for determining elastic continuum dynamic response using modal analysis

A modal analysis method which minimizes a weighted error function for the differential equation of an elastic continuum is presented. The technique essentially determines optimum time functions for the modal amplifiers by a simple extremization step. The method reduces to a differential equation collocation method when the error weighting function is a Dirac delta function over the continuum space. By means of an example, the technique is shown to have the potential advantages of modal decoupling despite modal non-orthogality, increased accuracy and simplified implementation compared to a more standard approach such as Lagrange's equations. The method is shown to be equivalent to Lagrange's equations for the special case of orthogonal modes.     

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.     

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.     

A theory for decoupling lossless coupled tapered lines with applications to directional couplers

A simple theory is presented for decoupling a pair of lossless coupled nonuniform transmission lines (CNUTL), with a common return and supporting TEM waves, into two lossless nonuniform transmission lines (NUTL). This theory is independent of the port terminations, symmetry conditions, etc. The method directly relates the line parameters of the lossless CNUTL's to those of the decoupled lines and vice versa; further, the matrix parameters of the CNUTL's as a four-port are explicitly expressed in terms of those of the decoupled lines as two-ports. This theory is then utilized to study the applications of CNUTL's as directional couplers.It is shown that for CNUTL's to behave as a codirectional coupler, each of the decoupled lines should be a proportional line, while for contradirectional coupler action, the two decoupled lines have to be duals of each other. The coupling response of the codirectional coupler is found to be periodic, while the phase shift between coupled and transmitted signals varies linearly with frequency. The coupling response of various contradirectional couplers with smooth transition at one of the ends, for which the decoupled lines are “basic NUTL's with hyperbolic solutions”, are studied in detail. It is shown that all these couplers have a high-pass response and that the CNUTL's with “hyperbolic cosine squared lines” as decoupled lines have the best response of all the CNUTL's considered.     

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.     

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.     

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 conductance of aqueous solutions of electrolytes

Since the Debye-Hückel theory clearly indicates, from thermodynamic considerations, that strong electrolytes are substantially completely dissociated in aqueous solutions, it has been necessary to replace, in the explanation of the conductances of such electrolytes, the Arrhenius theory, which assumes partial dissociation and constant ion mobilities, with a theory which provides for changing mobilities with increasing ion concentrations. To this end Debye and Hückel, and Onsager, suggest two mechanisms: the “electrophoretic effect” and “the time of relaxation effect,” both dependent upon interionic electrostatic attractions and repulsions. Experimental tests of Onsager's relations show that his equations account accurately, at least for very dilute solutions, for the changes with concentration of the conductances of solutions of electrolytes containing ions of widely differing mobilities and valencies. For higher concentrations semi-empirical equations are available. In contrast to the Arrhenius theory, the newer ideas call for changes in the transference numbers with the concentration, which have been found experimentally. Precision transference measurements have also furnished an experimental test of Kohlrausch's law of independent mobilities of ions. The theory of interionic attractions and repulsions is, in addition, of utility in dealing with the ionic part of solutions of weak electrolytes. By computing degrees of dissociation in a manner which allows for changing ion mobilities a test is obtained of the Debye-Hückel limiting relation connecting activity coefficients with ion concentrations.     

On the Computation of Functionals Over Sample Paths of a Randomly Rotating Rigid Body

This paper deals with the computation of the values of two functionals which are defined over the sample paths of a randomly rotating rigid body. It is assumed that the body is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the body's motion, two functionals are defined over its sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations is suggested, and its efficiency and applicability are demonstrated with examples.     

Variational analysis of electrical networks

A unified variational approach to network analysis is presented. From an initial classification of variables, expressed in topological terms, Lagrangian energy functions are generated which may be used in Hamilton's principles. This topological formalism allows extension into nonconservative networks and enables the complete retention of classical formalism in all networks.By using a topological version of the Brayton-Moser mixed potential function, several power variational principles are developed. These principles lead to comparison of the roles played by classical energy and power state functions in analytical mechanics.     

Least squares finite element approximation of Navier's equation

In this paper we construct a collection of vector functions which are later used as a basis in finding a least squares finite element approximation to Navier's equation. Through ideas from potential theory, pointwise convergence is demonstrated.     

An isoperimetric problem for continuous systems: The aeolian vibration of a conductor span

The equations of motion for perfectly mechanical systems which are restricted to satisfy given constraints are derived through the Lagrangian formulation for continuous media. Hamilton's equations for this situation are derived, and the equations for the balance of energy and angular momentum are generated from fundamental principles. These equations are all specialized to the example of a conductor which is vibrating under Aeolian excitation. Various models for the Hamiltonian density and the constraint energy density due to the wind energy input are also discussed.     

On the Salient Properties of the Hysteresis Machine—I. Non-conducting Rotor

The properties of the hysteresis machine are analyzed on the basis of a cylindrical model. The electromagnetic field in the various regions of the motor is derived by field-theoretical methods (such as Maxwell's equations and Poynting's theorem), and the resulting parameters are applied in turn in determining torques and other relevant quantities. Three limiting particular cases of special interest are examined.     

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.     

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.     

Elements of the First Column of Routh's Array

A simple explicit recursive formula for the elements of the first column of Routh's array is presented. The use of this formula will eliminate the need for constructing the whole array.     

On the Salient Properties of the Hysteresis Machine—II. Eddy-current Effects

The analysis of the asynchronous behavior of the hysteresis machine is undertaken based on a model describing its cylindrical structure. Assuming that the hysteresis shell of the rotor is manufactured out of materials with finite conductivities, the influence of the induced eddy- currents on the machine performance is pointed out via Maxwell's equations and Poynting's theorem. The characteristics of the machine are computed as functions of slip and three interesting limiting cases are revealed.     

Efflux of gases through orifices—kinetic theory

The efflux of gases through orifices was treated heretofore on basis of adiabatic expansion, postulating the decrease of temperature, which is not observed in the facts of everyday experience. The consideration of molecular motion on the principles of the Kinetic Theory of Gases shows the possibility of the expansion of a gas flowing through an orifice without change of temperature, and permits the formation of simple quantitative relations which are in a satisfactory agreement with test data. For small difference in pressure the new expression for discharge becomes identical with Fliegner's formula. It is explained that reduction of temperature postulated by adiabatic formulæ should be understood to apply to the observer moving with stream, and that no change of temperature is registered by stationary observer. The possibility of extension of the adiabatic and of the new “kinetic” formulæ to the pressure ratios below critical is opened by the consideration of reduced number of impacts between molecules with reduction of the pressure ratio.     

Eddy Current Effects on the Asynchronous Performance of the Hysteresis Machine

An analysis of the asynchronous behavior of the hysteresis machine is undertaken based on a “rectified” model of its structure. The hysteresis shell and the yoke of the rotor are assumed as being manufactured out of materials with finite conductivities. The field analysis undertaken via Maxwell's equations and Poynting's theorem shows the influence of the induced eddy currents on the machine performance at nonvanishing slips. The characteristics of the machine are computed as functions of slip, revealing three particularly interesting cases.