Linear Approximation Theory Research Articles

Active tendon control of cable‐stayed bridges: a large‐scale demonstration

AbstractThis paper presents a strategy for active damping of cable structures, using active tendons. The first part of the paper summarizes the theoretical background: the control law is briefly presented together with the main results of an approximate linear theory which allows the prediction of closed‐loop poles with a root locus technique. The second part of the paper reports on experimental results obtained with two test structures: the first one is a small size mock‐up representative of a cable‐stayed bridge during the construction phase. The control of the parametric vibration of passive cables due to deck vibration is demonstrated. The second one is a 30 m long mock‐up built on the reaction wall of the ELSA test facility at the JRC Ispra (Italy); this test structure is used to demonstrate the practical implementation of the control strategy with hydraulic actuators. Copyright © 2001 John Wiley & Sons, Ltd.

Active Tendon Control of Large Trusses

The use of tension cables to stiffen and control precision trusses as needed for future interferometric missions is investigated. A strategy for damping cable structures with active tendons is presented. Each tendon consists of a displacement actuator (piezoelectric in this case) collocated with a force sensor; the local control law consists of an integral force feedback, which has guaranteed stability if we assume perfect actuator and sensor dynamics. Then an approximate linear theory that allows one to predict the closed-loop poles of a cable structure with a root locus technique is developed; the methodology is applied numerically to a model of the Jet Propulsion Laboratory Micro-Precision Interferometer. Finally, a laboratory experiment on a guyed truss with three active tendons is described, and the experimental results are compared with the numerical predictions.

Active Tendon Control of Vibration of Truss Structures: Theory and Experiments

This paper proposes a strategy for the active damping of cable structures, using active tendons. The first part of the paper summarizes the theoretical background: the control law is first briefly presented and the main results of an approximate linear theory which allows to predict the closed-loop poles with a root-locus technique are mentioned. The second part of the paper reports on experimental results obtained with two truss structures representative of future space applications. Finally, the behaviour of the control system for microvibrations is investigated.

Numerical Analysis of Nonlinear Wave Propagation in Elastic Submerged Cables

This paper analyzes the coupled nonlinear tangential-normal waves that propagate along underwater cable suspensions. Taken with the recently developed linear theory governing the in-plane structural waves (Behbahani-Nejad and Perkins, 1996) and an analysis of nonlinear out-of-plane waves for submerged cables (Behbahani-Nejad and Perkins, 1997), this investigation contributes further understanding toward a nonlinear three-dimensional theory for wave propagation along fluid-loaded cables. The nonlinearities present in the in-plane model render the cable/fluid model intractable by exact analytical methods. A numerical solution is pursued in this study using finite difference algorithms. To this end, an infinite cable domain is divided to two subdomains, namely an interior (finite computational) domain and exterior (infinite far-field) domain. Closed-form solutions for the approximate linear theory are employed for the far field in constructing nonreflecting boundary conditions for the computational domain. Numerical results highlight the governing role of nonlinear hydrodynamic drag for underwater cable suspensions. The numerical results further demonstrate that most analyses of cables in air are not useful for underwater cable applications.

INFLUENCE OF AMPLIFIER AND FEEDBACK ON THE DYNAMICS OF A WATER HYDRAULIC SERVOVALVE

This paper deals with the influence of electronic elements, especially the saturation of servo-amplifier and the demodulator filter in the feedback loop of the servovalve. Nonlinear numerical analysis and linear approximation theory are applied to calculate the frequency responses and the wave forms of the system variables. The demodulator filter for LVDT results in an increase of gain and a decrease of a phase lag of spool motion. Thus, it can improve the system response under suitable combination of parameters. The saturation of servo-amplifier distorts wave forms of flapper and spool motion, and renders their frequency response worse.

Linear Least-Squares algorithms for temporal difference learning

We introduce two new temporal diffence (TD) algorithms based on the theory of linear least-squares function approximation. We define an algorithm we call Least-Squares TD (LS TD) for which we prove probability-one convergence when it is used with a function approximator linear in the adjustable parameters. We then define a recursive version of this algorithm, Recursive Least-Squares TD (RLS TD). Although these new TD algorithms require more computation per time-step than do Sutton's TD(λ) algorithms, they are more efficient in a statistical sense because they extract more information from training experiences. We describe a simulation experiment showing the substantial improvement in learning rate achieved by RLS TD in an example Markov prediction problem. To quantify this improvement, we introduce theTD error variance of a Markov chain, ωTD, and experimentally conclude that the convergence rate of a TD algorithm depends linearly on ωTD. In addition to converging more rapidly, LS TD and RLS TD do not have control parameters, such as a learning rate parameter, thus eliminating the possibility of achieving poor performance by an unlucky choice of parameters.

Monte Carlo modeling of underwater-image formation: validity of the linear and small-angle approximations

<p>A Monte Carlo model has been used to compute a set of point-spread functions (PSF's) and modulation transfer functions (MTF's) that determine underwater-image quality in a range of different environments. The results have been used to analyze the range of application under which a linear-approximation theory holds. Conclusions of the study are that the linear-approximation theory Seems to hold quite well over a broad range of applications. The ramifications of the Wells small-angle-scattering theory that predicts the PSF from a knowledge of the volume-scattering function (VSF) are also considered.</p><p>Discrepancies are noted between a predicted and a computationally obtained MTF; these discrepancies increase with range. Therefore, the results of the simulations indicate that the small-angle-scattering theory is more valid at a limited number of attenuation lengths. The results of the simulations indicate that the theory is valid to approximately three attenuation lengths.</p>

Post-Newtonian Equations of Motion in the Flat Universe

Using the (3 + 1) formalism, we derive the post-Newtonian (PN) equations of motion in a flat unvierse. To derive the equations of motion, we must carefully consider two points, one being the choice of the density in the Newtonian order (ρN) and the other the choice of the gauge condition. In choosing ρN, we require that the density fluctuation ρN - ρ0 agrees with a gauge invariant quantity in the linear approximation theory. As a gauge condition, we propose the cosmological post-Newtonian (CPN) slice condition with the pseudo transverse-traceless gauge condition, by which the evolution of the geometric variables derived in the PN approximation in the early stage of universe agrees with that of the gauge invariant quantities in the linear approximation. In the derived equations of motion, the force is calculated from six potentials which satisfy the Poisson equations. Hence, our formalism can be easily applied to numerical simulations in which the standard technique (e.g., particle-mesh method) is used. We apply the PN formula to the one-dimensional (1D) Zel'dovich solution to demonstrate that our strategy works well, and also to determine the effect of the PN forces on the evolution of the large-scale structure. It is found that the behavior of the density fluctuation and metric quantities in the early stage obtained by the present formalism agrees with that of the gauge invariant quantities in the linear approximation, although they do not always agree within the previous formalism due to the appearance of spurious gauge modes. We also discuss the evolution of the non-linear density fluctuation with very large scale, which may be affected by the PN correction in the last stage of the evolution.

Exact quantization condition for anharmonic oscillators (in one dimension)

An exact quantization condition is given for the one-dimensional Schrodinger operator with a homogeneous anharmonic potential q2M. It has the form of an explicit mapping from level sequences to level sequences, involving a Bohr-Sommerfeld-like quantization step, and having the exact spectrum as fixed point. Numerical tests and an approximate linear theory both suggest, at least for the few lowest M, that the mapping has a contractive region: when an initial level sequence is only asymptotically correct to lowest order, its iterates are seen to converge term by term towards the exact eigenvalues. This type of approach ought to extend to general polynomial potentials.

An algorithm for form error evaluation — using the theory of discrete and linear Chebyshev approximation

Abstract ISO defines form error based on the concept of minimum zone, but does not describe the methods to obtain the same. In this paper, drawbacks of the available methods are discussed and a new algorithm is presented. This algorithm is guaranteed to give the minimum value of form error. The algorithm is well-suited for implementation on a computer which may be interfaced with a Coordinate Measuring Machine.

Theories and error analysis of pencil tracing

To efficiently produce a more realistic image, the paraxial theory is introduced into computer graphics. The paraxial theory is a linear approximation theory established in other fields such as lens designing. The paraxial theory is formulated by a 4 × 4 matrix called “system matrix,” and this provides theoretical bases for various rendering methods such as pencil tracing, beam tracing, and cone tracing. This paper outlines the paraxial theory, analyzes errors from the viewpoint of its application to an image generation, and proposes a compact error-estimation function. The efficiency of the function is demonstrated experimentally. Basic data to develop an image-generation technology based on the paraxial theory are presented.

Base Isolation: Linear Theory and Design

The idea that a building can be uncoupled from the damaging effects of the ground movement produced by a strong earthquake has appealed to inventors and engineers for more than a century. Many ingenious devices have been proposed to achieve this result, but very few have been implemented and the concept now referred to as base isolation or seismic isolation has yet to be generally accepted by the engineering profession. Although most of the proposed systems are unacceptably complicated, in recent years a few practical systems have been developed and implemented. While some of these systems have been tested on large-scale shaking tables, none have to date been tested as-built by a strong earth tremor. The shake table testing and related static testing of full-scale components such as isolation bearings, however, has led to a certain degree of acceptance by the profession and it is possible that the number of practical implementations of base isolation will increase quite dramatically in the next few years. This paper describes recent implementations of base isolation and describes an approximate linear theory of isolation which can be used for the design of base isolation systems that use multilayer elastomeric isolators.

A Mechanical Approach to the Determination of the Geometry of a Woven Fabric and to the Analysis of Subsequent Changes in This Geometry Part I: A Theory for the Crimping of the Weft Yarn during Weaving

The problem of the weft yarn acquiring crimp during the process of weaving is simulated by the deformation of an elastic rod of known length, fixed at both ends and loaded by a point force in the middle. Two analyses, one based on large deformations, the other on an approximate linear theory, which take into account both bending and longitudinal extension, are carried out and tested experimentally on a physical-simulation model. It is also shown that both theories can be applied to real yarns.

A Mechanical Approach to the Determination of the Geometry of a Woven Fabric and to the Analysis of Subsequent Changes in This Geometry Part II: The Development of a Geometry of Plain-woven Fabric in the Loomstate and a Theory of Weaving Contractions

On the basis of theories developed in Part I of this series, it is shown that the geometry of a plain-woven cloth in its tight state on the loom can be calculated from the deformations of the warp and weft yarns in the process of weaving. Simulating the state of the grey cloth as the tension components of the forces on the loomstate fabric vanish, two theories of weaving contraction are developed. The approximate linear theory is tested experimentally on a physical model of the plain-weave cloth formed with rubber filaments. It is also shown by actual weaving experiments that both theories can be successfully applied to real fabrics.

A learning theorem for linear operators

We prove a convergence theorem in linear dynamic approximation theory which yields estimates of certain series of errors.

Spectral Estimators that Extend the Maximum Entropy and Maximum Likelihood Methods

The theory of best linear approximation in weighted $L^2 $ spaces is used to obtain a general procedure, the PDFT, for linearly reconstructing the Fourier transform from sampled data. The PDFT can be used either directly to reduce sidelobe structure and to extrapolate the data or indirectly to obtain high resolution spectral estimators. The direct and indirect PDFT include as special cases many of the commonly used spectral techniques, including Burg’s maximum entropy method, Capon’s maximum likelihood method, the spectral estimators based on bandlimited extrapolation, the eigenvalue/eigenvector methods for detecting sinusoids in noise (Pisarenko method, Schmidt’s MUSIC, eigenvector power beamforming), and the best linear unbiased estimator (BLUE) for regression coefficients. By exploiting their relationship to the linear PDFT, these nonlinear techniques can be analyzed in terms of linear approximation theory. In addition to providing a unifying formulation for many different spectral estimators, the PDFT...

Reliability Analysis of Structural Members Composed of Several Random Components, Making Use of Theory of Random Processes∗

Abstract This paper presents a method of reliability analysis of structural members, considering both the load and the resistance to be composed of several random components. The analysis is based on the theory of linear statistical approximation, extended to random processes. To illustrate application of the present theory, numerical calculations are carried out on a singly reinforced concrete beam with rectangular cross section and on a steel compression member with random initial deflections

Computation of all solutions to a system of polynomial equations

This paper proposes a homotopy continuation method for approximating all solutions to a system of polynomial equations in several complex variables. The method is based on piecewise linear approximation and complementarity theory. It utilizes a skilful artificial map and two copies of the triangulationJ 3 with continuous refinement of grid size to increase the computational efficiency and to avoid the necessity of determining the grid size a priori. Some computational results are also reported.

Linear dynamic approximation theory

We study in detail the behavior of some known learning algorithms. We estimate the sum of the squares of the absolute relative errors of some general linear learning algorithms and the sum of the squares of the coefficients obtained by the perceptron algorithm. We prove the convergence of a statistical learning algorithm. The possibility of applications of this theory to biology is discussed.

Some remarks on H-sets in linear approximation theory

Some remarks on H-sets in linear approximation theory