Research Papers

On Instability Pockets and Influence of Damping in Parametrically Excited Systems

[+] Author and Article Information
Ashu Sharma

Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: azs0111@auburn.edu

S. C. Sinha

Life Fellow ASME
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 6, 2017; final manuscript received February 15, 2018; published online March 30, 2018. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 140(5), 051001 (Mar 30, 2018) (9 pages) Paper No: VIB-17-1486; doi: 10.1115/1.4039406 History: Received November 06, 2017; Revised February 15, 2018

In most parametrically excited systems, stability boundaries cross each other at several points to form closed unstable subregions commonly known as “instability pockets.” The first aspect of this study explores some general characteristics of these instability pockets and their structural modifications in the parametric space as damping is induced in the system. Second, the possible destabilization of undamped systems due to addition of damping in parametrically excited systems has been investigated. The study is restricted to single degree-of-freedom systems that can be modeled by Hill and quasi-periodic (QP) Hill equations. Three typical cases of Hill equation, e.g., Mathieu, Meissner, and three-frequency Hill equations, are analyzed. State transition matrices of these equations are computed symbolically/analytically over a wide range of system parameters and instability pockets are observed in the stability diagrams of Meissner, three-frequency Hill, and QP Hill equations. Locations of the intersections of stability boundaries (commonly known as coexistence points) are determined using the property that two linearly independent solutions coexist at these intersections. For Meissner equation, with a square wave coefficient, analytical expressions are constructed to compute the number and locations of the instability pockets. In the second part of the study, the symbolic/analytic forms of state transition matrices are used to compute the minimum values of damping coefficients required for instability pockets to vanish from the parametric space. The phenomenon of destabilization due to damping, previously observed in systems with two degrees-of-freedom or higher, is also demonstrated in systems with one degree-of-freedom.

Copyright © 2018 by ASME
Topics: Stability , Damping , Waves
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Fig. 1

Stability diagram of damped Mathieu equation (10) with d=0 and 0.5

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Fig. 2

Stability diagram of damped Mathieu equation (10) in the a∼d plane (b=0.5)

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Fig. 3

Stability diagram of damped Mathieu equation (10) with d=0 and 0.5

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Fig. 4

Critical values of damping for points B and C

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Fig. 5

Unit rectangular waveform coefficient in the damped Meissner equation

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Fig. 6

Stability diagrams of Meissner equation for c=0.3  and  0.5

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Fig. 7

Effect of damping on instability pockets of Meissner equation with a square wave coefficient

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Fig. 8

Effect of damping on instability pocket PM5

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Fig. 9

Destabilizing effect of damping in Meissner equation (c=0.5)

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Fig. 10

Stability diagram of three-frequency Hill equation (23) with δ=1

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Fig. 11

Stability diagram of three-frequency Hill equation (23) with δ=0.53 and δ=0.6

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Fig. 12

Stability diagram of QP Hill equation with ω1=1.0 and ω2=(1+5)/2. Solid: 2T periodic; dashed: T periodic.




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