Modeling Of Flexible Structures Research Articles

柔軟宇宙構造物に対するＨ∞制御における最悪な外乱について （第１報） 理論的考察

Objective of the present study is to clarify the characteristics of the worst disturbance on flexible space structures. “The worst disturbance” is defined as a disturbance which most effectively causes the vibration of a dynamical system and is formulated theoretically by using the idea of H∞ control theory and the optimal control problem. Flexible structures are modeled by an Euler-Bernoulli beam and the worst disturbance on the flexible space structure model is obtained mathematically by numerical simulation.

Cross-Section Structure of Cylindrical and Polymer-like Micelles from Small-Angle Scattering Data. 2. Experimental Results

We report a small-angle neutron scattering (SANS) study of the cross-section structure of polymer-like lecithin reverse micelles in deuterated cyclohexane. We demonstrate that the application of the indirect Fourier transformation and square-root deconvolution methods to data from SANS measurements with cylindrical polymer-like micelles allows for a direct verification of the previously postulated geometrical model of flexible tubular structures with a well defined water core and a surfactant shell. By combining contrast variation experiments and data analysis performed on an absolute scale, we quantitatively deduce information on properties such as the extension of the aqueous core and the degree of water penetration into the headgroup and solvent penetration into the tail region.

Modelling of an Elastic Disk With Finite Hub Motions and Small Elastic Vibrations With Application to Rotordynamics

In the modelling of flexible structures undergoing large overall motion and small elastic vibrations, it is necessary to include elastic displacement effects up to second order in the kinematical equations if the resulting differential equations are to be consistent. Elastic displacements are typically modelled using mode shapes and generalized coordinates, and second and higher order elastic effects (sometimes referred to as foreshortening) are often neglected in the development of kinematical relations. This study examines the implications of such effects in the modelling of an elastic disk with arbitrary base motions and small elastic vibrations. A general modelling procedure is described that is appropriate for the development of simulation models for such structures. An approximate technique is used to account for second order elastic terms in the kinematical relations. The modelling procedure is specialized to the case of an elastic disk undergoing a constant axial spin and infinitesimal displacements for all other degrees of freedom. Comparisons are made between the natural frequencies of a disk model with and without these elastic foreshortening effects and some conclusions are drawn as to the relative importance of such terms in rotor disk modelling.

Generalization of the Wittrick-Williams formula for counting modes of flexible structures

A recent (1971) result due to Wittrick-Williams relating the number of modes of a flexible structure to the zeros of a matrix function is generalized and an independent proof given. It is applied to Timoshenko models of a class of interconnected beamlike lattice structures and an interpretation of the zeros given as pinned modes of the structure. N determining modes of continuum models of flexible structures, particularly when the mode density is high, it is often necessary to make sure that no modes have been missed. Here, a result due to Wittrick and Williams1 (which is at once an algorithm and often referred to as such) may be used to determine the total number of modes in any given interval of frequency. In the words of the authors, it is an algorithm in the sense that it enables one to calculate how many natural frequencies lie below any chosen frequency, without determining them and hence to converge to any required natural frequency to any specified accuracy. References 2-5 are illustrative of the actual use of this algorithm for determining modes of beamlike structures. Our concern in this paper is in the formula itself rather than its use as a computational tool for finding modes of structures. Indeed, it can be regarded as a generalization of Foster's theorem for onedimensional loss-less impedances, familiar in classical electrical circuit theory; see Ref. 6. Also, the proof given by the authors in Ref. 6 leans heavily on a classical theorem of Lord Rayleigh7 and many arguments peculiar to mechanics of structures. Here we formulate the problem in the generality of an abstract wave equation in a Hilbert space and at the same time present an entirely independent mathematical proof. We show, in addition, how it can be applied to Timoshenko models of interconnected beamlike lattice structures.

Eigenvalue perturbation models for robust control

This paper presents a nonconservative description of the regions containing the eigenvalues of a diagonal matrix, perturbed by the set of all unknown, Euclidean norm bounded matrices. This is extended to real valued block diagonal matrices where each two-by-two block reveals a complex conjugate pair of eigenvalues. A weighting matrix allows one to specify the exact size of the perturbation to each of the eigenvalues. An identical result is obtained for real valued perturbations. This result is motivated by, and applied to, the modeling of frequency and damping perturbations in models of flexible structures. The resulting perturbation description fits within the established H/sub /spl infin// robust control framework and, in certain situations, is less conservative than the more standard additive or multiplicative perturbation models. >

Parameter identification on active control of a structural model

The dynamic characteristics of a flexible 3DOF structural model, namely the stiffness and damping ratios, are identified by using the extended Kalman filtering technique. Two types of structural response datum are used for the identification. These are the structural response under base excitation generated by a shaking table and the forced vibration response due to an active mass damper (AMD) installed at the top of the model. Using the identified structural parameters, experimental results of the earthquake response of the model with the AMD control are compared with those of analytical simulation. Fairly good agreement between the experiment and the numerical simulation suggests the accuracy of the identification and the effectiveness of the vibration control by the AMD.

Modeling and Vibration Control of Flexible Structures Arranged in Parallel.

This paper proposes the vibration control method of flexible structures arranged in parallel. In recent years, vibration control of a flexible structures using an active dynamic absorber has increasingly been utilized in practical applications. In the future, as the structure becomes more advanced and more flexible, its natural frequency gradually becomes lower. Thus, it will be difficult to assume enough force for vibration control using an active dynamic absorber. The authors proposed the method wherein two structures arranged in parallel are controlled actively using two actuators set between them. In general, a reduced-order-model is necessary to design a controller using the LQ control theory. Using the modeling method proposed here, reduced-order model of the 2-2 degree-of-freedom parallel structures can be easily fabricated. Using a root locus plot, the relationship between weighting parameters and control effect in the LQ control theory is clarified. It is demonstrated that this control method is able to control vibrations of 2-2 DOF system without the spillover instability, by simulation and experiment.

A New Method for Making a Reduced-Order Model of Flexible Structures using Unobservability and Uncontrollability and Its Application in Vibration Control

This paper proposes a new method for modeling flexible structures with distributed parameters as reduced-order models with lumped parameters. Both prevention of spillover and physical correspondence at the modeling points are taken into consideration. When modern control theory is employed to control the vibration of flexible structures, because of restrictions on controller design, it is necessary to make a reduced-order model of the structure. When higher modes are ignored, the reduction of the model may result in vibration instability-spillover. Because of uncontrollability and unobservability, nodes of higher-order vibration modes are selected as modeling points to prevent spillover. The effctiveness of this method is demonstrated by applying vibration control to a flexible towerlike structure.

Approximate methods for dynamic response of multi-module floating structures

Two-dimensional and three-dimensional approximate hydroelasticity theories are used to predict the dynamic response of 5- and 16-module very large floating structures in regular waves. For the two-dimensional approach, strip theory is combined with a beam model of the structure to determine the dynamic response. However, the fluid forces from strip theory are augmented with ‘surge’ forces calculated from Morison's equation. For the three-dimensional approach, a rigid module, flexible connector structural model is used with fluid loading obtained with the three-dimensional double composite source distribution technique. Fluid-structure interaction effects are studied and connector moments are obtained. These approximate methods of hydroelastic analysis offer alternatives to the more computationally demanding, fully three-dimensional flexible module, flexible connector model in which linear potential theory is used.

Zeros of modal models of flexible structures

It is shown that in some situations the zero patterns of a flexible structure change radically with model order. The refinement of the model by the addition of higher-order modes can cause zeros to appear at low frequencies or in the right half-plane. Structures instrumented with piezoelectric actuators and/or fiber-optic sensors are also considered. >

Orthogonal canonical forms for second-order systems

It is shown that a linear second-order system with arbitrary damping cannot be reduced to Hessenberg-triangular form by means of orthogonal transformations. However, it is also shown that such an orthogonal reduction is always possible for the modal damping commonly assumed for models of flexible structures. It is shown that modally damped models can be orthogonally reduced to a new triangular second-order Schur form.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Time-domain structural identification using free response measurements

The problem of determining linear time-invariant models of flexible structures is considered. The approach is first to identify the modal parameters by using free response measurement records that could be contaminated with noise. The stiffness and damping matrices are then extracted from the identified modal parameters. The identification technique, which is developed in the time-domain is based on least square linear regression theory and is organized as a systematic and easy to implement scheme. The estimates are produced without the need of solving an eigenvalue problem or singular value decomposition of a data matrix. The effect of measurement noise on estimation accuracy is discussed. A numerical example is presented to illustrate the results.

Likelihood estimation for distributed parameter models of large beam-like structures

In this paper, a maximum likelihood estimation for distributed parameter models of large flexible structures has been formulated. Distributed parameter models involve far fewer unknown parameters than independent modal characteristics or finite element models. The closed form solutions for the partial differential equations with corresponding boundary conditions have been derived. The closed form expressions of the sensitivity functions lead to highly efficient algorithms for analyzing ground or on-orbit test results. For illustration of this approach, experimental data of the NASA Mini-Mast truss have been used. The estimations of modal properties involve its lateral bending modes and torsional modes. The results show that distributed parameter models are promising in the parameter estimation of large flexible structures.

Robust control with respect to coprime factors of infinite-dimensional positive real systems

It is shown that infinite-dimensional positive real systems can be robustly stabilized with respect to coprime factor perturbations with a robustness margin of at least 1/ square root 2. This result is applied to dissipative colocated systems which arise in PDE models of flexible structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On existence and uniqueness of solutions of a class of perturbed second order evolution equations

In this paper, properties of operators of the kind A + Λ are investigated, where A is a selfadjoint, positive definite differential operator with compact inverse on a Hilbert space, and Λ is assumed to be A-bounded, A-symmetric and A-positive semidefinite. A general theory on the existence and uniqueness of solutions of second order evolution equations involving the operator A + Λ is developed. The new findings obtained in this paper have potential applications in modeling and control of flexible structures.

Mission function control for a slew maneuver experiment

A control algorithm named mission function control is experimentally demonstrated and verified on a slew maneuver of a flexible space structure model. The mission function control algorithm employs a Lyapunov-type function that consists of generalized energy functions. The model consists of a rigid body having a cantilevered flexible appendage; it is controlled to slew in a horizontal plane by a torque motor. Analytical study indicates that vibrational motion of the flexible appendage can be sensed by strain gauges as a bending moment and shear force at the root of the appendage. Results of the experiment show that a simple implementation of the algorithm leads to excellent controlled behavior of the slew maneuver as well as excellent control robustness.

Modal truncation, Ritz vectors, and derivatives of closed-loop damping ratios

The effect of modal truncation on the damping ratio and their derivatives with respect to an added mass is investigated for a simply supported, multispan beam with a linear quadratic Gaussian control system. It is found that both the damping ratios and derivatives converge slowly, but the derivatives converge more slowly than the damping ratios. However, it is shown that when Ritz vectors corresponding to static displacements due to actuator forces are added to the reduced model, the convergence of both the damping ratios and their derivatives is accelerated. It is also shown that the accuracy of the damping ratio predicted by a reduced-model control design can be improved significantly if the Ritz vectors are included in the design of the control system. Thus, it appears that Ritz vectors added to the reduced model of flexible structures can improve greatly the accuracy of both the design and analysis of the control system.

Transfer function analysis of a flexible toroidal structure

This paper is concerned with the modeling of a flexible toroidal structure for attitude and structural control studies. A two-dimensiona l elastic system, consisting of a flexible toroid and a pretensioned membrane, is analyzed in terms of the transfer function from a control torque input to the collocated sensor output. The critical tension for out-of-plane buckling is also determined. The models considered in the paper are simple enough to treat analytically, yet complicated enough to demonstrate the dynamical characteristics of future space structures with a toroidal configuration. It is shown that for such structures, the coupling between two elastic systems results in a transfer function expressed as a combination of an infinite product expansion and an infinite partial fraction expansion. Such a representation should be useful in future studies in reduced-order modeling of flexible toroidal structures.

Identification of flexible structures for robust control

Documentation is provided of the authors' experience with modeling and identification of an experimental flexible structure for the purpose of control design, with the primary aim being to motivate some important research directions in this area. A multi-input/multi-output (MIMO) model of the structure is generated using the finite element method. This model is inadequate for control design, due to its large variation from the experimental data. Chebyshev polynomials are employed to fit the data with single-input/multi-output (SIMO) transfer function models. Combining these SIMO models leads to a MIMO model with more modes than the original finite element model. To find a physically motivated model, an ad hoc model reduction technique which uses a priori knowledge of the structure is developed. The ad hoc approach is compared with balanced realization model reduction to determine its benefits. Descriptions of the errors between the model and experimental data are formulated for robust control design. Plots of select transfer function models and experimental data are included. >

Modeling, Control, and Design of Flexible Structures: A Survey

Large structures used in space applications, are made flexible to reduce the high cost of lifting mass to orbit. These highly flexible structures will inevitably be vulnerable to dynamic excitations caused by the slewing/pointing maneuvers, thermal transients, operations of on-board coolers and generators, etc. Thus the aspects of modeling and control become extremely important for the safe and effective operation of large flexible structures (LFS’s). The theoretical, computational, and practical methodologies developed during the last decade along with the latest trends in this area are described. The research work addressing the aspects of model reduction, passive and active control, hierarchical/decentralized control, integrated structural and control design, optimal design, and sensor/actuator location selection problem for large flexible structures is discussed.