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Research Papers

Conditions of Parametric Resonance in Periodically Time-Variant Systems With Distributed Parameters

[+] Author and Article Information
Yong-Kwan Lee1

Central R&D Institute, SAMSUNG Electro-Mechanics Co., Maetan 3-Dong, Suwon, Gyungi-Do 443-743, Republic of Korealurpy@hanmail.net

Leonid S. Chechurin

Department of Innovation Theory, St. Petersburg State Polytechnic University, 195251 Polytechnicheskaya str., 29 St. Petersburg Russiacepreu4@gmail.com

1

Corresponding author.

J. Dyn. Sys., Meas., Control 131(3), 031008 (Mar 20, 2009) (11 pages) doi:10.1115/1.3072151 History: Received November 06, 2007; Revised November 08, 2008; Published March 20, 2009

Theoretical analysis of the stability problem for the control systems with distributed parameters shall be given. The approach to the analysis of such systems can be composed of two parts. First, the distributed parameter element is modeled by a frequency response function. Second, approximate conditions of parametric resonance are derived by a method of stationarization (describing functions of time-variant elements). The approach is illustrated by two examples. One is a robot-manipulator arm (distributed mechanical parameter system) controlled by a controller with a modulator/demodulator cascade (time-varying element). Another is an electromechanical transformer that consists of a constant current motor and a synchronous generator. Inductance between stator windings and the rotor of the synchronous generator serves as a periodical time-varying parameter, and a long electrical line plays the role of an element with distributed parameters. In both examples, dangerous (in terms of the first parametric resonance) regions for time-varying parameter are obtained theoretically and compared with simulation experiment.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Block diagram of the time-variant system

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Figure 2

First parametric resonance circle

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Figure 3

Parametric resonance excitation condition

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Figure 4

Illustration of instability analysis for periodical time-variant systems with distributed parameters in complex (a) Nyquist and (b) inverse Nyquist planes

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Figure 5

Equivalent representation of a time-variant system

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Figure 6

Central parametric circle with frequency response of systems with distributed parameters in (a) complex plane and (b) frequency-magnitude plane

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Figure 7

Example 1: modified block scheme with centered time-variant element

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Figure 8

An Illustration of parametric resonance conditions for a time-variant system with distributed parameters in (a) a Nyquist plane and (b) a magnitude-phase plane

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Figure 9

Example 2: robot-manipulator system

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Figure 10

Transient response of stationary lumped system with compensation

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Figure 11

Inverse Nyquist plot of the servo system with distributed parameters and parametric circumference

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Figure 12

Schematic for the numerical modeling of the total system with a time-varying element

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Figure 13

Time histories of the shaft: (a) transient signal in the absence of time-varying parameter, (b) general view of the transient signal in the presence of the time-varying parameter, and (c) zoomed view of (b)

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Figure 14

Example 3: motor-generator electrical transformer system

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Figure 15

Alternating character of the mutual inductance according to time during shaft rotation

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Figure 16

Conditions of first parametric resonance excitement on electromechanical transformer with long line (250 m) loading: (a) Nyquist diagram and (b) magnitude versus frequency plot

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Figure 17

Parametric excitation according to the length of the electrical line: dotted line, rotation speed of the shaft (300 rad/s)

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Figure 18

Block scheme of SIMULINK for numerical experiment of parametric excitation about the electromechanical transformer

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Figure 19

Oscillation growth for the electromechanical transformer by numerical experiment. (a) General view and (b) fragment of (a) (dotted line, variation of the time-variant parameter; solid line, oscillation in the system).

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