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

Subhash C. Sinha

Alumni Professor Emeritus Life Fellow, ASME Department of Mechanical Engineering, Auburn University, Auburn, AL 36849

1Corresponding author.

ASME 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 sub-regions 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. Secondly, the possible destabilization of undamped systems due to addition of damping in parametrically excited systems has been investigated. The study is restricted to SDF systems that can be modeled by Hill and Quasi-Periodic 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 Quasi-Periodic 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 (c) 2018 by ASME
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