Journal ArticleDOI
Perturbation solutions of the Carson–Cambi equation
Barbara Epstein,Richard Barakat +1 more
TLDR
In this paper, the Carson-Cambi equation (1+e cos t)y + py = 0, referred to as the second-order differential equation with a periodic coefficient associated with the second derivative was examined.Abstract:
The periodic differential equation (1+e cos t)y + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |e|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–e plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to e3. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±12, ±32, ±52. Three-timing is employed to extend the solution near k = ±12. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.read more
Citations
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Journal ArticleDOI
Stability analysis of systems with periodic coefficients: An approximate approach
S.C. Sinha,C.C. Chou,H.H. Denman +2 more
TL;DR: In this paper, an approximate method of stability analysis for second order linear systems with periodic coefficients is presented, where the periodic functions are approximated during the first period of motion by a constant, a linear or a quadratic function of time such that the resulting approximate equations have known closed form solutions.
Journal ArticleDOI
A Quantitative Stability Analysis of the Solutions to the Carson-Cambi Equation
TL;DR: In this paper, a quantitative stability analysis of the Carson-Cambi equation is carried through, using a new, effective approach, and the results are compared with a recent perturbation analysis, and show that this should not be used for e 0.4.
Book ChapterDOI
Sensitivity Analysis for Dynamic Stability Problems
TL;DR: In this paper, the authors present a set of six notes written to be fairly independent without extensive mutual reference and the page limit for each note is set to 10. This page limit means that extensive examples are not included.
Journal ArticleDOI
The random capacitance equation
TL;DR: In this article, the initial value problem for a circuit containing a small random capacitance is studied and the method of first-order smoothing is employed to determine the mean solution.
Journal ArticleDOI
Numerical Simulation of Stability and Responses of Dynamic Systems under Parametric Excitation
TL;DR: In this paper , the authors proposed a numerical simulation method to simultaneously construct stability diagrams and predict the responses of multiple degree-of-freedom (DOF) dynamic systems under arbitrary parametric loadings.
References
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