TY - JOUR
T1 - A Floquet-Based Analysis of Parametric Excitation through the Damping Coefficient
AU - Afzali, Fatemeh
AU - Acar, Gizem D.
AU - Feeny, Brian F.
N1 - Publisher Copyright:
© 2021 American Society of Mechanical Engineers (ASME). All rights reserved.
PY - 2021/8
Y1 - 2021/8
N2 - The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu's equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.
AB - The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu's equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.
KW - Coexistence
KW - Damped oscillator
KW - Damping
KW - Floquet theory
KW - Harmonic balance
KW - Parametric excitation
KW - Response characteristics
KW - Stability
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U2 - 10.1115/1.4048392
DO - 10.1115/1.4048392
M3 - Article
AN - SCOPUS:85107912087
SN - 1048-9002
VL - 143
JO - Journal of Vibration and Acoustics, Transactions of the ASME
JF - Journal of Vibration and Acoustics, Transactions of the ASME
IS - 4
M1 - 041003
ER -