Transfer Function Formulation of Constrained Distributed Systems

1991 ◽  
Author(s):  
C. A. Tan ◽  
C. H. Chung
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
Constrained Systems ◽  
Natural Boundary ◽  
Mode Localization ◽  
Function Formulation ◽  
General Displacement ◽  
Curve Veering ◽  
Natural Boundary Conditions ◽  
Pointwise Constraints

Abstract The transfer function formulation of constrained distributed systems is presented. The methodology is illustrated for subsystems under pointwise constraints and distributed systems consisting of multiple subsystems. A general displacement method (GDM) is used to determine the eigensolution of the constrained systems. It is shown that GDM requires only the natural boundary conditions (force constraints) be imposed at the subsystem interface. The methodology is applied to two examples. Curve veering and mode localization phenomena are found in an elastic structure on an elastic foundation.

Transfer Function Analysis of Non-Uniformly Distributed Parameter Systems

1993 ◽  
Author(s):  
Bingen Yang ◽  
Houfei Fang
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
State Space ◽  
Fundamental Matrix ◽  
Function Analysis ◽  
Distributed Parameter Systems ◽  
System Response ◽  
Distributed Parameter ◽  
Transfer Function Analysis ◽  
Function Formulation

Abstract This paper studies a transfer function formulation for general one-dimensional, non-uniformly distributed systems subject to arbitrary boundary conditions and external disturbances. The purpose is to provide an useful alternative for modeling and analysis of distributed parameter systems. In the development, the system equations of the non-uniform system are cast into a state space form in the Laplace transform domain. The system response and distributed transfer functions are derived in term of the fundamental matrix of the state space equation. Two approximate methods for evaluating the fundamental matrix are proposed. With the transfer function formulation, various dynamics and control problems for the non-uniformly distributed system can be conveniently addressed. The transfer function analysis is also applied to constrained/combined non-uniformly distributed systems.

Eigenvalue Inclusion Principles for Distributed Gyroscopic Systems

1992 ◽  
Vol 59 (3) ◽  
pp. 650-656 ◽  
Author(s):  
B. Yang
Keyword(s):  
Transfer Function ◽  
Discrete System ◽  
Natural Frequencies ◽  
Gyroscopic System ◽  
Positive Real ◽  
Constrained System ◽  
Function Formulation ◽  
Pointwise Constraints ◽  
Gyroscopic Systems ◽  
Vibrating Systems

In his famous treatise The Theory of Sound, Rayleigh enunciated an eigenvalue inclusion principle for the discrete, self-adjoint vibrating system under a constraint. According to this principle, the natural frequencies of the discrete system without and with the constraint are alternately located along the positive real axis. Although it is commonly believed that the same rule also applied for distributed vibrating systems, no proof has been given for the distributed gyroscopic system. This paper presents several eigenvalue inclusion principles for a class of distributed gyroscopic systems under pointwise constraints. A transfer function formulation is proposed to describe the constrained system. Five types of nondissipative constraints and their effects on the system natural frequencies are studied. It is shown that the transfer function formulation is a systematic and convenient way to handle constraint problems for the distributed gyroscopic system.

Transfer Function Formulation of Constrained Distributed Parameter Systems, Part II: Applications

1993 ◽  
Vol 60 (4) ◽  
pp. 1012-1019 ◽  
Author(s):  
C. H. Chung ◽  
C. A. Tan
Keyword(s):  
Transfer Function ◽  
Elastic Foundation ◽  
Normal Modes ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Displacement Method ◽  
Mode Localization ◽  
Function Formulation ◽  
Free Beam ◽  
Symmetric Systems

In this paper, the application of the transfer function formulation and the generalized displacement method (GDM) to the analysis of constrained distributed parameter systems is illustrated. Two kinds of classical examples are considered. In the constrained free-free beam example, it is shown how the GDM gives the eigensolutions without requiring knowledge of the normal modes of the unconstrained beam. In the string on a partial elastic foundation example, mode localization and eigenvalue loci veering phenomena are examined. It is shown that mode localizaation can occur in spatially symmetric systems and for modes whose frequency loci do not veer.

Eigenvalue Inclusion Principles for Distributed Gyroscopic Systems

1991 ◽  
Author(s):  
B. Yang
Keyword(s):  
Transfer Function ◽  
Natural Frequencies ◽  
Gyroscopic System ◽  
Constrained System ◽  
Function Formulation ◽  
Pointwise Constraints ◽  
Gyroscopic Systems

Abstract This paper presents several eigenvalue inclusion principles for a class of distributed gyroscopic systems under pointwise constraints. A transfer function formulation is proposed to describe the constrained system. Five types of non-dissipative constraints and their effects on the system natural frequencies are studied It is shown that the natural frequencies of the constrained gyroscopic system alternate with those of the unconstrained system.

Transfer Function Formulation of Constrained Distributed Parameter Systems, Part I: Theory

1993 ◽  
Vol 60 (4) ◽  
pp. 1004-1011 ◽  
Author(s):  
C. A. Tan ◽  
C. H. Chung
Keyword(s):  
Transfer Function ◽  
Distributed Parameter Systems ◽  
Constrained Systems ◽  
Distributed Parameter ◽  
Finite Dimensional ◽  
Function Formulation ◽  
Coordinate Vector ◽  
Generalized Displacements ◽  
Generalized Force ◽  
Forced Responses

Analysis of constrained distributed parameter systems by the transfer function formulation is presented. The methodology is suitable for symbolic computation coding. The distributed system is an assembly of distributed elements. A generalized displacement method (GDM) is developed to evaluate the free and forced responses of the system. It is shown that, while classical methods require the satisfaction of both the displacement and force boundary conditions at the subsystem interfaces, GDM only needs to impose generalized force constraints. The continuity of generalized displacements at the interfaces is embedded in the present formulation. Thus, computation efforts are greatly reduced, in particular, for systems with a large number of distributed subsystems. Eigenfunctions of constrained systems are obtained by solving a finite-dimensional eigenvalue problem governing the generalized coordinate vector. The formulation can be applied to damped and non-selfadjoint systems.

scholarly journals Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka B. Malinowska ◽  
Delfim F. M. Torres
Keyword(s):  
Fractional Integral ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Fractional Integrals ◽  
Natural Boundary ◽  
Free Boundary Value Problems ◽  
Fractional Variational Problems ◽  
Generalized Fractional Integral ◽  
Fractional Calculus Of Variations ◽  
Natural Boundary Conditions

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

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