Journal
JOURNAL OF VIBRATION AND ACOUSTICS-TRANSACTIONS OF THE ASME
Volume 138, Issue 4, Pages -Publisher
ASME
DOI: 10.1115/1.4033341
Keywords
Mathieu equation; Floquet theory; harmonic balance
Categories
Funding
- National Science Foundation [CMMI-1335177]
- Div Of Civil, Mechanical, & Manufact Inn
- Directorate For Engineering [1335177] Funding Source: National Science Foundation
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Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.
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