Two-end controllability of elastic vibrations of systems with distributed and lumped parameters

Boundary control problems for the vibrations of a system with distributed and lumped parameters are solved. The vibrations of a distributed-parameter object are described by boundary value problems with boundary conditions of various types. A lumped-parameter object is described by a second-order ordinary differential equation.     

Neural net versus classical models for the detection and localization of leaks in pipelines

Four models of a pipeline are compared in the paper: a nonlinear distributed-parameter model, a linear distributed-parameter model, a simplified lumped-parameter model and an extended neural-net-based model. The transcendental transfer function of the linearized model is obtained by a Laplace transformation and corresponding initial and boundary conditions. The lumped-parameter model is obtained by a Taylor series extension of the transencdental transfer function. Based on the experience of linear models the structure of the neural net model, as an addendum to the nonlinear distributed-parameter model, is obtained. All four models are tested on a real pipeline data with an artificially generated leak.     

Control optimization in a hyperbolic linear system

An optimal control problem is considered for a distributed-parameter Goursat-Darboux system with controlled boundary conditions. For the numerical solution of the problem, an algorithm based on separability and minimax theorems is constructed, which reduces the problem to finding the maximum of a concave functional defined in the class of one-variable functions.     

Solving an optimal control problem for parabolic equations

We consider optimal boundary control of a distributed-parameter system. The system state is described by two parabolic equations of second order, where the coefficients of one equation depend on the gradient of the solution of the second equation. An existence and uniqueness theorem is proved for the optimal control in this problem and the necessary conditions of optimality are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 90–98, 1986.     

A matrix Green's formula and optimal control of linear distributed-parameter systems

A matrix Green's formula is derived, and its application to the optimal control problem of distributed-parameter systems with both distributed and boundary inputs is presented.     

Asymptotic Analysis of State Constrained Semilinear Optimal Control Problems

The asymptotic behavior of state-constrained semilinear optimal control problems for distributed-parameter systems with variable compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit. We gratefully acknowledge the support of the DAAD. The paper was prepared during the visit of the first author at the Institute of Applied Mathematics II, University Erlangen-Nuremberg in 2003.     

On an optimal control problem for a hyperbolic partial differential equation

Existence of solutions is proved for a minimum problem for a distributed-parameter control system described by a linear, hyperbolic partial differential equation. The cost function is an integral depending on boundary controls.This research was supported by the Consiglio Nazionale delle Ricerche, Rome, Italy.     

Extremum principles for optimal control in normed linear spaces

Extremum principles intended for use in optimal control are derived in the form of necessary conditions and sufficient conditions, formulated in general normed linear spaces. The method of application is illustrated by several examples involving optimal control problems, mathematical programming problems, lumped-parameter systems, and distributed-parameter systems. The basic theorems provide a unified approach which is applicable to a wide variety of problems in open-loop optimal control.     

Extensions of the generalized moment theorem and applications to distributed-parameter control systems

In this paper, we present extensions to the generalized moment theorem and apply it to optimal control problems for a certain class of distributed-parameter systems. We also apply it to the time-optimal control problem and extend the results of Ref. 1 pertaining to the largest controllable set, so that we can discuss the problem of recoverability for some distributed-parameter systems.The author wishes to express his gratitude to Professor P. K. C. Wang for his guidance and suggestions.     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

Exact boundary controllability of partial nodal profile for network of strings

Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls, and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce the concept of in-situ controlled node to deal with a new situation that one node can be charged and controlled simultaneously. The minimum number of boundary controls on the entire tree-like network is determined by using the concept of ‘degree of freedom of charged nodes’ introduced. And the concept of ‘control path’ is introduced to appropriately divide the network, so that we can determine the infimum of controllability time. General frameworks of constructive proof are given on a single interval, a star-like network, a chain-like network and a planar tree-like network for linear wave equation(s) with Dirichlet, Neumann, Robin and dissipative boundary conditions to establish a complete theory on the exact boundary controllability of partial nodal profile.     

Nonlinear vibration of moderately thick laminated beams using finite element method

A finite element model is developed to study the large-amplitude free vibrations of generally-layered laminated composite beams. The Poisson effect, which is often neglected, is included in the laminated beam constitutive equation. The large deformation is accounted for by using von Karman strains and the transverse shear deformation is incorporated using a higher order theory. The beam element has eight degrees of freedom with the inplane displacement, transverse displacement, bending slope and bending rotation as the variables at each node. The direct iteration method is used to solve the nonlinear equations which are evaluated at the point of reversal of motion. The influence of boundary conditions, beam geometries, Poisson effect, and ply orientations on the nonlinear frequencies and mode shapes are demonstrated.     

Generalized thermoelasticity of beams under partial thermal shock

In this paper, the generalized thermoelastic response of a beam subjected to a partial lateral thermal shock is analysed. The beam is made of homogeneous and isotropic material and is assumed to follow the Hooke law for its constitutive material. The displacement gradient is small and the linear form of strain-displacement relations is used for the beam. The equations of motion and the boundary conditions of the beam are derived based on Hamilton’s principle. According to the first and second laws of thermodynamics, a non-Fourier constitutive equation is employed to derive the energy equation of the beam. The non-Fourier effects lead to the constitutive equation of the hyperbolic type and thus the thermal and mechanical waves can be observed. The propagation of waves in the beam are simulated by finite element model and the wave reflections for different types of boundary conditions are studied. The relaxation time is considered as a significant parameter and results show that energy absorption of the structure and the wave propagation speed depend upon this parameter.     

A new Orthonormal Polynomial Series Expansion Method in vibration analysis of thin beams with non-uniform thickness

In this article, OPSEM (Orthonormal Polynomial Series Expansion Method) is developed as a new computational approach for the evaluation of thin beams of variable thickness transverse vibration. Capability of the OPSEM in assessing the free vibration frequencies and mode shapes of an Euler–Bernoulli beam with varying thickness is discussed. Multispan continuous beams with various classical boundary conditions are included. Contribution of BOPs (Basic Orthonormal Polynomials) in capturing the beam vibrations is also illustrated in numerical examples to give a quantitative measure of convergence rate. Furthermore, OPSEM is adopted for the forced vibration of a thin beam caused by a moving mass. Dynamics of beams supported by flexible elastic base like free to free beam on elastic foundation are also regarded. Verifications are made via eigenfunction expansion method and GMLSM (Generalized Moving Least Square Method). The very close observed agreement between the results of the two recently mentioned methods and that of OPSEM can be regarded as a guarantee of validity for the newly introduced technique. In comparison with eigenfunction expansion method, the simplicity and handiness of OPSEM in coping with different boundary conditions of the beam can be considered as its benefit for engineering practitioners.     

Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation

Vibration of non-uniform beams with different boundary conditions subjected to a moving mass is investigated. The beam is modeled using Euler–Bernoulli beam theory. Applying the method of eigenfunction expansion, equation of motion has been transformed into a number of coupled linear time-varying ordinary differential equations. In non-uniform beams, the exact vibration functions do not exist and in order to solve these equations using eigenfunction expansion method, an adequate set of functions must be selected as the assumed vibration modes. A set of polynomial functions called as beam characteristic polynomials, which is constructed by considering beam boundary conditions, have been used along with the vibration functions of the equivalent uniform beam with similar boundary conditions, as the assumed vibration functions. Orthogonal polynomials which are generated by utilizing a Gram–Schmidt process are also used, and results of their application show no advantage over the set of simple non-orthogonal polynomials. In the numerical examples, both natural frequencies and forced vibration of three different non-uniform beams with different shapes and boundary conditions are scrutinized.     

Free vibration analysis of a rigid bar supported by arbitrary elastic beams

For convenience, a two-node conventional elastic beam element (C beam element) with the displacements of its 2nd node replaced by those of center of gravity (c.g.) of the joined rigid bar is called the modified beam element (M beam element). The objective of this paper is to present a modified finite element method (modified FEM) such that the free vibration characteristics of a rigid bar supported by a number of elastic beams can be easily determined. First of all, the displacements for the 2nd node of a C beam element joined with the rigid bar are determined in terms of those for the c.g. of the joined rigid bar to establish the M beam element. Next, the mass and stiffness matrices for the M beam element are derived based on the displacements for the 1st node of the C beam element and those for the c.g. of the joined rigid bar. Then, the overall property matrices of the entire unconstrained vibrating system (i.e. a rigid bar supported by a number of elastic beams) can be determined by using the assembly technique of the conventional FEM and considering the effects of lumped mass and rotary inertia of the rigid bar. Finally, the boundary (supporting) conditions are imposed to produce the effective property matrices of the constrained vibrating system and then the free vibration characteristics are determined with the standard approach. In order to confirm the presented theory and the developed computer program, the rigid bar is modeled by a number of C beam elements with bigger Young’s modulus (E_R) and the conventional FEM is used to determine the natural frequencies and associated mode shapes of the vibrating system. It is found that the latter will converge to the corresponding ones obtained from the presented modified FEM when the magnitude of E_R increases to certain values.     

Effect of damping on observation spillover instability

The control of a distributed-parameter system can be effected by a transformation to a finite-dimensional discrete system in terms of modal coordinates. If only a limited number of modal coordinates is controlled, then a phenomenon that has come to be known as control and observation spillover occurs. Observation spillover has been demonstrated to cause instability in undamped systems. This paper shows that a minimal amount of damping can eliminate the instability, at least for the case considered.This work was supported by the Naval Research Laboratory, Space Systems Division, Advanced Systems Branch, under ONR Research Grant No. N00014-78-C-0194.     

非均质Euler-Bernoulli梁的边界反馈镇定问题

基于频域乘子方法，讨论非均质Euler-Bernoulli梁边界反馈镇定问题，利用黄发伦关于C0-半群指数稳定的判据和指数乘子技巧，证明了只用力或力矩反馈，由Euler-Bernoulli梁所决定的闭环系统可以指数稳定。     

Observability of elastic vibrations of a beam

The observability problem for beam vibrations described by a fourth-order partial differential equation with various boundary conditions is considered. Dynamic observability problems are solved in terms of boundary conditions and observations of the beam state at certain fixed instants of time.