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.

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.

A Nonlinear Oscillator Lanchester Damper

1987 ◽  
Vol 109 (4) ◽  
pp. 343-347 ◽  
Author(s):  
K. R. Asfar ◽  
A. H. Nayfeh ◽  
K. A. Barrash
Keyword(s):  
Steady State ◽  
Nonlinear Oscillator ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Harmonic Excitation ◽  
Second Order ◽  
Numerical Results ◽  
Nonlinear Spring ◽  
Uniform Expansion ◽  
Steady State Solutions

The method of multiple scales is used to investigate the effect of a nonlinear spring in the main system on the performance of Lanchester-type absorbers. A second-order uniform expansion is obtained for the response of the system to a harmonic excitation. Numerical results for steady-state solutions illustrating the influence of the nonlinearity and damping factors on the response are presented. A softening-type effective nonlinearity dominates the system and considerably improves its damping.

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.

scholarly journals Perturbations of nonlinear autonomous oscillators

1994 ◽  
Vol 35 (4) ◽  
pp. 445-468 ◽  
Author(s):  
P. B. Chapman
Keyword(s):  
General Theory ◽  
Fourier Coefficients ◽  
Multiple Scales ◽  
Second Order ◽  
Multiple Scales Method ◽  
Suitable Parameter ◽  
Non Linear

AbstractA general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

On the solution of Hill's equation using Milne's method

1991 ◽  
Vol 24 (9) ◽  
pp. 2069-2081 ◽  
Author(s):  
J A Nunez ◽  
F Bensch ◽  
H J Korsch
Keyword(s):  
Hill’S Equation ◽  
Hill's Equation

Vibration and Stability of an Elastic Beam Subjected to a Periodic Axial Load With Time-Dependent Displacement Excitation at Both Ends

1991 ◽  
Author(s):  
Xiao-Feng Wu ◽  
Adnan Akay
Keyword(s):  
Axial Load ◽  
Perturbation Technique ◽  
Harmonic Balance Method ◽  
Elastic Beam ◽  
Time Dependent ◽  
Transverse Load ◽  
Order Partial Differential Equation ◽  
Weighted Residual Method ◽  
Hill’S Equation ◽  
Hill's Equation

Abstract This paper concerns the transverse vibrations and stabilities of an elastic beam simultaneously subjected to a periodic axial load, a distributed transverse load, and time-dependent displacement excitations at both ends. The equation of motion derived from Bernoulli-Euler beam theory is a fourth-order partial differential equation with periodic coefficients. To obtain approximate solutions, the method of assumed-modes is used. The unknown time-dependent function in the assumed-modes method is determined by a generalized inhomogeneous Hill’s equation. The instability regions possessed by this generalized Hill’s equation are obtained by both the perturbation technique up to the second order and the harmonic balance method. The dynamic response and the corresponding spectrum of the transversely oscillating elastic beam are calculated by the weighted-residual method.

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