### Abstract

The minimum damping for asymptotic stability is predicted for Hill’s equation with any bounded parametric excitation. It is shown that the response of Hill’s equation with bounded parametric excitation is exponentially bounded. The parametric excitation maximizing the bounding exponent is identified by time optimal control theory. This maximal bounding exponent is balanced by viscous damping to ensure asymptotic stability. The minimum damping ratio is calculated as a function of the excitation bound. A closed form, more conservative estimate of the minimum damping ratio is also predicted. Thus, if the general (e.g., unknown, aperiodic, orrandom) parametric excitation of Hill’s equation is bounded, a simple, conservative estimate of the damping required for asymptotic stability is given in this paper.

Original language | English (US) |
---|---|

Pages (from-to) | 366-370 |

Number of pages | 5 |

Journal | Journal of Applied Mechanics, Transactions ASME |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1993 |

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### All Science Journal Classification (ASJC) codes

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

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*Journal of Applied Mechanics, Transactions ASME*, vol. 60, no. 2, pp. 366-370. https://doi.org/10.1115/1.2900802

**On the stability of the damped hill’s equation with arbitrary, bounded parametric excitation.** / Rahn, Christopher D.; Mote, C. D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the stability of the damped hill’s equation with arbitrary, bounded parametric excitation

AU - Rahn, Christopher D.

AU - Mote, C. D.

PY - 1993/1/1

Y1 - 1993/1/1

N2 - The minimum damping for asymptotic stability is predicted for Hill’s equation with any bounded parametric excitation. It is shown that the response of Hill’s equation with bounded parametric excitation is exponentially bounded. The parametric excitation maximizing the bounding exponent is identified by time optimal control theory. This maximal bounding exponent is balanced by viscous damping to ensure asymptotic stability. The minimum damping ratio is calculated as a function of the excitation bound. A closed form, more conservative estimate of the minimum damping ratio is also predicted. Thus, if the general (e.g., unknown, aperiodic, orrandom) parametric excitation of Hill’s equation is bounded, a simple, conservative estimate of the damping required for asymptotic stability is given in this paper.

AB - The minimum damping for asymptotic stability is predicted for Hill’s equation with any bounded parametric excitation. It is shown that the response of Hill’s equation with bounded parametric excitation is exponentially bounded. The parametric excitation maximizing the bounding exponent is identified by time optimal control theory. This maximal bounding exponent is balanced by viscous damping to ensure asymptotic stability. The minimum damping ratio is calculated as a function of the excitation bound. A closed form, more conservative estimate of the minimum damping ratio is also predicted. Thus, if the general (e.g., unknown, aperiodic, orrandom) parametric excitation of Hill’s equation is bounded, a simple, conservative estimate of the damping required for asymptotic stability is given in this paper.

UR - http://www.scopus.com/inward/record.url?scp=0027607025&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027607025&partnerID=8YFLogxK

U2 - 10.1115/1.2900802

DO - 10.1115/1.2900802

M3 - Article

VL - 60

SP - 366

EP - 370

JO - Journal of Applied Mechanics, Transactions ASME

JF - Journal of Applied Mechanics, Transactions ASME

SN - 0021-8936

IS - 2

ER -