The Stability Surfaces of a Hill’s Equation With Several Small Parameters

1973 ◽  
Vol 40 (4) ◽  
pp. 1107-1109
Author(s):  
L. A. Rubenfeld
Keyword(s):  
Perturbation Method ◽  
Second Order ◽  
Hill’S Equation ◽  
Hill's Equation ◽  
Small Parameters ◽  
The Stability

The stability surfaces of a Hill’s equation with three independent small parameters are investigated using an extension of a perturbation method due to Struble. The method is applied up to second order but it is clear that it can be extended to any desired order and to other equations having any number of small parameters.

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.

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.

Periodic solutions of a linear differential equation of the second order with periodic coefficients

1926 ◽  
Vol 23 (1) ◽  
pp. 44-46 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Periodic Solutions ◽  
Second Order ◽  
Hill’S Equation ◽  
Periodic Coefficients ◽  
Hill's Equation ◽  
Linear Differential ◽  
Image Position

The equation to be considered is of the typewhere p (x) is continuous for all real values of x, even, and periodic. It is no restriction to suppose that the period is π, and this assumption will be made, so that the equation is virtually Hill's equation.

Characteristic Exponents and Stability of Hill’s Equation

1972 ◽  
Vol 39 (4) ◽  
pp. 1156-1158 ◽  
Author(s):  
Ali Hasan Nayfeh
Keyword(s):  
Multiple Scales ◽  
Second Order ◽  
Hill’S Equation ◽  
Characteristic Exponents ◽  
Hill's Equation ◽  
Uniform Expansion ◽  
Transition Curves

The methods of strained parameters and multiple scales are used to determine a second-order uniform expansion for the solutions of Hill’s equation near the transition curves. Second-order explicit expansions are presented for the characteristic exponents and the transition curves.

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.

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