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Research Papers

Closed Form Control Gains for Zero Assignment in the Time Delayed System

[+] Author and Article Information
Kumar Vikram Singh1

Department of Mechanical and Manufacturing Engineering, Miami University, Oxford, OH 45056singhkv@muohio.edu

Biswa Nath Datta

Department of Mathematical Sciences, Northern Illinois University, De Kalb, IL 60115

Mayank Tyagi

Department of Petroleum Engineering, Louisiana State University, Baton Rouge, LA 70803

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(2), 021002 (Oct 20, 2010) (9 pages) doi:10.1115/1.4002340 History: Received September 30, 2009; Revised January 04, 2010; Published October 20, 2010; Online October 20, 2010

Control of the vibrating structures is desirable in various engineering applications for preventing fatigue and failure. It can be achieved by passive means using dynamic absorbers or by active means using sensors and actuators. In some cases, it is also not practical to apply a desirable control force in those locations at which the dynamics of the structure are to be controlled. In recent years, dynamic absorption schemes are investigated in which control strategies that absorb a steady state motion of a desired location in the structure have been developed. Such a vibration control strategy is termed as zero assignment. Unlike conventional full-state feedback control, which requires all the states of the system to be measured, zero assignment requires least numbers of sensors and actuators (depending on the number of dynamic absorption points) for estimating the control gains and, hence, it may provide economical engineering solution. However, while applying control strategy by active zero assignment, small time delay from the sensors and actuators in the feedback loop is unavoidable and they influence the control gains as well as the stability of the system. In this paper, we have developed vibration control strategy by active zero assignment and obtained closed form control gains for systems with and without time delays by using truncated and full Taylor series expansion. Some examples related to conservative and nonconservative systems as well as realistic distributed parameter systems are presented to demonstrate the active dynamic absorption and the effects of time delay on control gains. The effect of delay in the stability of the controlled system is also summarized.

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Figures

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Figure 1

Schematic of zero assignment strategy for the aircraft wing

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Figure 2

A harmonically excited nonconservative multidegree of freedom system

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Figure 3

(a) Uncontrolled system, (b) controlled system: control applied at the third degree of freedom, and (c) controlled system: control applied at the second degree of freedom

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Figure 4

The total response of the fourth degree of freedom

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Figure 5

Comparison of the control gains obtained from truncated and full Taylor series

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Figure 6

(a) An axially vibrating fixed-free uniform rod and (b) the associated discrete mass-spring model of order n obtained by finite difference method

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Figure 7

Control gains α(ω) and β(ω) for suppressing the axial vibration at the free end of the rod by applying the control at xc (a) and (b) truncated Taylor series and (c) and (d) full Taylor series

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Figure 8

The variation in the pole closest to the imaginary axis as defined in Eq. 51 with respect to the time delay (a) excitation frequency ω=3 and (b) excitation frequency ω=3

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Figure 9

The variation in the pole closest to the imaginary axis as defined in Eq. 51 with respect to the time delay (a) excitation frequency ω=3 and (b) excitation frequency ω=3

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