Paraxial Maxwell beams: transformation by general linear optical systems

Extending the work of earlier papers on the relativistic-front description of paraxial optics and the formulation of Fourier optics for vector waves consistent with the Maxwell equations, we generalize the Jones calculus of axial plane waves to describe the action of the most general linear optical system on paraxial Maxwell fields. Several examples are worked out, and in each case it is shown that the formalism leads to physically correct results. The importance of retaining the small components of the field vectors along the axis of the system for a consistent description is emphasized.

A rational approach to the synthesis of one-dimensional acoustic filters

The velocity ratio algorithm developed from a heuristic study of transfer matrix multiplication has been employed to bring out the relative effects of the elements constituting a linear, one-dimensional acoustic filter, the overall dimensions of which are fixed, and synthesize a suitable straight-through configuration for a low-pass, wide-band, non-dissipative acoustic filter. The potential of the foregoing approach in applications to the rational design of practical acoustic filters such as automotive mufflers is indicated.

A note on the period of oscillation of non-linear systems

This paper deals with two approximate methods of finding the period of oscillations of non-linear conservative systems excited by step functions. The first method is an extension of the analysis presented by Jonckheere [4] and the second one is based on a weighted bilinear approximation of the non-linear characteristic. An example is presented and the approximate results are compared with the exact results

The transient response of certain third-ordernon-linear systems

In this paper a method of solving certain third-order non-linear systems by using themethod of ultraspherical polynomial approximation is proposed. By using the method of variation of parameters the third-order equation is reduced to three partial differential equations. Instead of being averaged over a cycle, the non-linear functions are expanded in ultraspherical polynomials and with only the constant term retained, the equations are solved. The results of the procedure are compared with the numerical solutions obtained on a digital computer. A degenerate third-order system is also considered and results obtained for the above system are compared with numerical results obtained on the digital computer. There is good agreement between the results obtained by the proposed method and the numerical solution obtained on digital computer.

Active vibration suppression of non-linear beams using optimal dynamic inversion

Euler–Bernoulli beams are distributed parameter systems that are governed by a non-linear partial differential equation (PDE) of motion. This paper presents a vibration control approach for such beams that directly utilizes the non-linear PDE of motion, and hence, it is free from approximation errors (such as model reduction, linearization etc.). Two state feedback controllers are presented based on a newly developed optimal dynamic inversion technique which leads to closed-form solutions for the control variable. In one formulation a continuous controller structure is assumed in the spatial domain, whereas in the other approach it is assumed that the control force is applied through a finite number of discrete actuators located at predefined discrete locations in the spatial domain. An implicit finite difference technique with unconditional stability has been used to solve the PDE with control actions. Numerical simulation studies show that the beam vibration can effectively be decreased using either of the two formulations.

Stabilization of linear and nonlinear dynamical systems using an observer-controller configuration

In this paper the problem of stabilization of systems by means of stable compensations is considered, and results are derived for systems using observer�controller structures, for systems using a cascade structure, and for nonlinear systems

On the bottleneck linear programming problem

In this paper, the results on primal methods for Bottleneck Linear Programming (BLP) problem are briefly surveyed, the primal method is presented and the degenerate case related to Bottleneck Transportation Problem (BTP) is explicitly considered. The algorithm is based on the idea of using auxiliary coefficients as is done by Garfinkel and Rao [6]. The modification presented for the BTP rectifies the defect in Hammer's method in the case of degenerate basic feasible solution. Illustrative numerical examples are also given.

On various types of decoupling for time-varying system

This paper deals with the problem of decoupling a class of linear time-varying multi-variable systems, based on the defining property that the impulse response matrix of a decoupled system is diagonal. Depending on the properties of the coefficient matrices of the vector differential equation of the open-loop system, the system may be uniformly or totally decoupled. The necessary and sufficient conditions that permit a system to be uniformly or totally decoupled by state variable feedback are given. The main contribution of this paper is the precise definition of these two classes of decoupling and a rigorous derivation of the necessary and sufficient conditions which show the necessity of requiring that the system be of constant ordered rank with respect to observability. A simple example illustrates the importance of having several definitions of decoupling. Finally, the results are specialized to the case of time invariant systems.

Comments on a decentralized control of linear multivariable systems

A strongly connected decentralized control system may be made single channel controllable and observable with respect to any channel by decentralized feedbacks. It is noted here that the system example considered by Corfmat and Morse to illustrate this fact is already single channel controllable and observable, with respect to one of the channels. An alternate example which fits into the situation is presented in this item.

A note on non-separable solutions of linear partial differential equation

A method has been presented for constructing non-separable solutions of homogeneous linear partial differential equations of the type F(D, D′)W = 0, where D = ∂/∂x, D′ = ∂/∂y, Image where crs are constants and n stands for the order of the equation. The method has also been extended for equations of the form Φ(D, D′, D″)W = 0, where D = ∂/∂x, D′ = ∂/∂y, D″ = ∂/∂z and Image As illustration, the method has been applied to obtain nonseparable solutions of the two and three dimensional Helmholtz equations.

Application of ultraspherical polynomials to forced oscillations of a third-order non-linear system

In this paper the method of ultraspherical polynomial approximation is applied to study the steady-state response in forced oscillations of a third-order non-linear system. The non-linear function is expanded in ultraspherical polynomials and the expansion is restricted to the linear term. The equation for the response curve is obtained by using the linearized equation and the results are presented graphically. The agreement between the approximate solution and the analog computer solution is satisfactory. The problem of stability is not dealt with in this paper.

Numerical studies on quasilinear and linear elliptic equations

We explore here the acceleration of convergence of iterative methods for the solution of a class of quasilinear and linear algebraic equations. The specific systems are the finite difference form of the Navier-Stokes equations and the energy equation for recirculating flows. The acceleration procedures considered are: the successive over relaxation scheme; several implicit methods; and a second-order procedure. A new implicit method—the alternating direction line iterative method—is proposed in this paper. The method combines the advantages of the line successive over relaxation and alternating direction implicit methods. The various methods are tested for their computational economy and accuracy on a typical recirculating flow situation. The numerical experiments show that the alternating direction line iterative method is the most economical method of solving the Navier-Stokes equations for all Reynolds numbers in the laminar regime. The usual ADI method is shown to be not so attractive for large Reynolds numbers because of the loss of diagonal dominance. This loss can however be restored by a suitable choice of the relaxation parameter, but at the cost of accuracy. The accuracy of the new procedure is comparable to that of the well-tested successive overrelaxation method and to the available results in the literature. The second-order procedure turns out to be the most efficient method for the solution of the linear energy equation.

Non-linear bending of beams of variable cross-section

For the non-linear bending of cantilever beams of variable cross-section, the effect of large deformations, but with linear elasticity, is considered. The governing integral equation is solved by a numerical iterative procedure. Results for some typical cases are obtained and compared with some of those available in the literature.

A new approach for fixed point smoothing in linear distributed parameter systems

In this paper, we solve the distributed parameter fixed point smoothing problem by formulating it as an extended linear filtering problem and show that these results coincide with those obtained in the literature using the forward innovations method.

Role of Capping Ligands on the Nanoparticles in the Modulation of Properties of a Hybrid Matrix of Nanoparticles in a 2D Film and in a Supramolecular Organogel

We incorporate various gold nanoparticles (AuNPs) capped with different ligands in two-dimensional films and three-dimensional aggregates derived from N-stearoyl-L-alanine and N-lauroyl-L-alanine, respectively. The assemblies of N-stearoyl-L-alanine afforded stable films at the air-water interface. More compact assemblies were formed upon incorporation of AuNPs in the air-water interface of N-stearoyl-L-alanine. We then examined the effects of incorporation of various AuNPs functionalized with different capping ligands in three-dimensional assemblies of N-lauroyl-L-alanine, a compound that formed a gel in hydrocarbons. The profound influence of nanoparticle incorporation into physical gels was evident from evaluation of various microscopic and bulk properties. The interaction of AuNPs with the gelator assembly was found to depend critically on the capping ligands protecting the Au surface of the gold nanoparticles. Transmission electron microscopy (TEM) showed a long-range directional assembly of certain AuNPs along the gel fibers. Scanning electron microscopy (SEM) images of the freeze-dried gels and nanocomposites indicate that the morphological transformation in the composite microstructures depends significantly on the capping agent of the nanoparticles. Differential scanning calorimetry (DSC) showed that gel formation from sol occurred at a lower temperature upon incorporation of AuNPs having capping ligands that were able to align and noncovalently interact with the gel fibers. Rheological studies indicate that the gel-nanoparticle composites exhibit significantly greater viscoelasticity compared to the native gel alone when the capping ligands are able to interact through interdigitation into the gelator assembly. Thus, it was possible to define a clear relationship between the materials and the molecular-level properties by means of manipulation of the information inscribed on the NP surface.