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.

On the Stability of Hill’s Equation With Four Independent Parameters

1969 ◽  
Vol 36 (4) ◽  
pp. 885-886 ◽  
Author(s):  
Richard H. Rand
Keyword(s):  
Floquet Theory ◽  
Hill’S Equation ◽  
Stability Problems ◽  
Hill's Equation ◽  
The Stability ◽  
Independent Parameters

The stability of Hill’s equation with four independent parameters is studied by using Floquet theory and perturbations. Examples are given which demonstrate how the resulting analysis may be applied to a wide variety of stability problems.

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.

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.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

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.

Sign in / Sign up

Export Citation Format

Share Document