On the Stability of the Damped Hill’s Equation With Arbitrary, Bounded Parametric Excitation

1993 ◽  
Vol 60 (2) ◽  
pp. 366-370 ◽  
Author(s):  
C. D. Rahn ◽  
C. D. Mote
Keyword(s):  
Asymptotic Stability ◽  
Parametric Excitation ◽  
Optimal Control Theory ◽  
Damping Ratio ◽  
Viscous Damping ◽  
Conservative Estimate ◽  
Hill’S Equation ◽  
Hill's Equation ◽  
Time Optimal ◽  
The Stability

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, or random) parametric excitation of Hill’s equation is bounded, a simple, conservative estimate of the damping required for asymptotic stability is given in this paper.

The stability of thermal oscillations in a reactive slab: reduction to Hill’s equation

1982 ◽  
Vol 384 (1787) ◽  
pp. 455-462
Keyword(s):  
Surface Temperature ◽  
Hill’S Equation ◽  
Periodic Surface ◽  
Temperature Oscillations ◽  
Hill's Equation ◽  
Chemically Reacting ◽  
The Stability ◽  
Thermal Oscillations ◽  
Surface Temperature Variation ◽  
Time Periodic

The thermal stability of an exothermic chemically reacting slab with time-periodic surface temperature variation is examined. It is shown, on the basis of a good approximation due to Boddington, Gray and Walker, that the behaviour depends on the solutions of an ordinary differential equation of first order. The equation contains a modified amplitude, for small values of which it can be reduced to a particular form of Hill’s equation. Critical values of the Frank-Kamenetskii parameter, as a function of the amplitude ϵ and frequency ω of the surface temperature oscillations, are derived from the latter equation. For ω = 2π and 0 ≼ ϵ ≼ 2 the values are in good agreement with previously calculated ones.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

Stability of General Hill’s Equation With Three Independent Parameters

1972 ◽  
Vol 39 (1) ◽  
pp. 276-278 ◽  
Author(s):  
K. Hamer ◽  
M. R. Smith
Keyword(s):  
Perturbation Method ◽  
New Method ◽  
Order Of Approximation ◽  
Hill’S Equation ◽  
Stability Boundaries ◽  
First Order ◽  
Hill's Equation ◽  
The Stability ◽  
Independent Parameters

The stability of Hill’s equation with three independent parameters, two of which are small, is analyzed using a perturbation method. It is shown that, except for periodic terms of a special type, existing methods of determining stability boundaries fail. A new method, which works successfully to the first order of approximation, is described.

Gravity-Gradient Excitation of a Rotating Cable-Counterweight Space Station in Orbit

1963 ◽  
Vol 30 (4) ◽  
pp. 547-554 ◽  
Author(s):  
V. Chobotov
Keyword(s):  
Parametric Excitation ◽  
Equations Of Motion ◽  
Space Station ◽  
Type A ◽  
Gravity Gradient ◽  
Viscous Damping ◽  
Stability Criteria ◽  
Axial Vibrations ◽  
The Stability ◽  
Transverse Oscillations

The gravity-gradient excitation of a whirling cable-counterweight space station in orbit is investigated. The Lagrange’s equations of motion for transverse oscillations of the system are derived and shown to be of the Mathieu type. A few representative cases are investigated analytically and on an IBM 7090 computer. The stability criteria for the axial vibrations are also considered and shown to be of the same kind as those for the transver se vibrations of the cable. Viscous damping is included in the analyses and found to be effective and essential for prevention of parametric excitation instability of the system.

Computing the stability regions of Hill's equation

2005 ◽  
Vol 162 (2) ◽  
pp. 639-660 ◽  
Author(s):  
Svetlana V. Simakhina ◽  
Charles Tier
Keyword(s):  
Hill’S Equation ◽  
Stability Regions ◽  
Hill's Equation ◽  
The Stability
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