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.

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.

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.

Active Mode Localization in Distributed Parameter Systems with Consideration of Limited Actuator Placement, Part 1: Theory

1995 ◽  
Vol 122 (2) ◽  
pp. 160-164 ◽  
Author(s):  
Franz J. Shelley ◽  
William W. Clark
Keyword(s):  
Distributed Parameter Systems ◽  
Theoretical Development ◽  
Distributed Parameter ◽  
Limiting Factor ◽  
Sensors And Actuators ◽  
Mode Localization ◽  
Active Mode ◽  
Value Decomposition ◽  
Actuator Placement ◽  
Do So

The purpose of this two-part work is to apply active mode localization to distributed parameter systems where the number of control sensors and actuators is a limiting factor. In this part, the theoretical development portion of the study, two approaches are presented that shape system eigenvectors using feedback control, generating localization to produce areas of isolation with relatively low vibration amplitudes compared to other parts of the structure. The first approach uniformly shapes all eigenvectors of a vibrating system, but can require many actuators to do so. The second more general approach uses singular value decomposition (SVD) to shape selected eigenvectors of a system, localizing the response of these modes to any disturbance, and requiring few actuators. [S0739-3717(00)70202-9]

Transfer Functions of One-Dimensional Distributed Parameter Systems by Wave Approach

2006 ◽  
Vol 129 (2) ◽  
pp. 193-201 ◽  
Author(s):  
B. Kang
Keyword(s):  
Transfer Function ◽  
Frequency Response ◽  
Wave Reflection ◽  
Transfer Functions ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Complex Frequency ◽  
Reflection And Transmission ◽  
One Dimensional ◽  
Analysis Technique

An alternative analysis technique, which does not require eigensolutions as a priori, for the dynamic response solutions, in terms of the transfer function, of one-dimensional distributed parameter systems with arbitrary supporting conditions, is presented. The technique is based on the fact that the dynamic displacement of any point in a waveguide can be determined by superimposing the amplitudes of the wave components traveling along the waveguide, where the wave numbers of the constituent waves are defined in the Laplace domain instead of the frequency domain. The spatial amplitude variations of individual waves are represented by the field transfer matrix and the distortions of the wave amplitudes at point discontinuities due to constraints or boundaries are described by the wave reflection and transmission matrices. Combining these matrices in a progressive manner along the waveguide using the concepts of generalized wave reflection and transmission matrices leads to the exact transfer function of a complex distributed parameter system subjected to an externally applied force. The transient response solution can be obtained through the Laplace inversion using the fixed Talbot method. The exact frequency response solution, which includes infinite normal modes of the system, can be obtained in terms of the complex frequency response function from the system’s transfer function. This wave-based analysis technique is applicable to any one-dimensional viscoelastic structure (strings, axial rods, torsional bar, and beams), in particular systems with multiple point discontinuities such as viscoelastic supports, attached mass, and geometric/material property changes. In this paper, the proposed approach is applied to the flexural vibration analysis of a classical Euler–Bernoulli beam with multiple spans to demonstrate its systematic and recursive formulation technique.

Transfer Functions of One-Dimensional Distributed Parameter Systems

1992 ◽  
Vol 59 (4) ◽  
pp. 1009-1014 ◽  
Author(s):  
B. Yang ◽  
C. A. Tan
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Distributed Parameter Systems ◽  
Axially Moving Beam ◽  
System Response ◽  
Distributed Parameter ◽  
One Dimensional ◽  
Adjoint Systems ◽  
The Laplace Transform ◽  
Axially Moving

Distributed parameter systems describe many important physical processes. The transfer function of a distributed parameter system contains all information required to predict the system spectrum, the system response under any initial and external disturbances, and the stability of the system response. This paper presents a new method for evaluating transfer functions for a class of one-dimensional distributed parameter systems. The system equations are cast into a matrix form in the Laplace transform domain. Through determination of a fundamental matrix, the system transfer function is precisely evaluated in closed form. The method proposed is valid for both self-adjoint and non-self-adjoint systems, and is extremely convenient in computer coding. The method is applied to a damped, axially moving beam with different boundary conditions.

Active Mode Localization in Distributed Parameter Systems with Consideration of Limited Actuator Placement, Part 2: Simulations and Experiments

1995 ◽  
Vol 122 (2) ◽  
pp. 165-168 ◽  
Author(s):  
F. J. Shelley ◽  
W. W. Clark
Keyword(s):  
Vibration Isolation ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Limiting Factor ◽  
Parameter System ◽  
Sensors And Actuators ◽  
Mode Localization ◽  
Active Mode ◽  
Shaping Technique ◽  
Actuator Placement

The purpose of this two-part work is to apply active mode localization techniques to distributed parameter systems where control actuator and sensor placement is a limiting factor. In this paper, Part 2 of the study, the SVD eigenvector shaping technique examined in Part 1 is utilized to numerically and experimentally localize the response of a simply supported beam. This is done for two reasons. First, it demonstrates the application of this modified mode localization technique to a distributed parameter system. Second, it shows that it is possible to use this method to produce vibration isolation, reducing the absolute displacements in designated portions of the system while simultaneously curtailing the number of necessary control sensors and actuators. [S0739-3717(00)70302-3]

The Normal Modes of Vibration of Certain Nonlinear Continuous Systems

1964 ◽  
Vol 31 (1) ◽  
pp. 139-140 ◽  
Author(s):  
Thein Wah
Keyword(s):  
Normal Modes ◽  
Distributed Parameter Systems ◽  
Structural Theory ◽  
Distributed Parameter ◽  
Continuous Systems ◽  
Simple Interpretation ◽  
Modes Of Vibration

Two examples from structural theory are invoked to show that “normal modes” of vibration could have a simple interpretation in nonlinear distributed parameter systems.

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.

The Nature of the Temporal Solutions of Damped Distributed Systems With Classical Normal Modes

1982 ◽  
Vol 49 (4) ◽  
pp. 867-870 ◽  
Author(s):  
D. J. Inman ◽  
A. N. Andry
Keyword(s):  
Distributed Systems ◽  
Normal Modes ◽  
Distributed Parameter Systems ◽  
Time Response ◽  
Distributed Parameter

Conditions under which the time response of certain distributed parameter systems, which are assumed to possess “classical normal” modes, is critically damped, overdamped, or underdamped are presented. The conditions are derived from the definiteness of certain combinations of the coefficient operators of the describing equations. These conditions are compared to previous results and their usefulness is illustrated by examples.

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