Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation

Ramani, D. V.; Keith, William L.; Rand, Richard H.

doi:10.1016/s0020-7462(02)00218-4

Cited by 35 publications

(11 citation statements)

References 2 publications

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“…(5) Then the solution d(t) can be composed piecewise from the solutions of (4) and (5). It is worth noting that both (4) and (5) are nonlinear Mathieu-like differential equations [5], [13].…”

Section: Model Of the Dynamicsmentioning

confidence: 99%

“…From a mathematical point of view, the simple equations that describe the taut-slack phenomenon pertain to the class of the nonlinear Mathieu equations, whose solutions can reveal a diversity of qualitative changes, see for instance analysis of bifurcations in quadratically-and cubically-damped Mathieu equation in [5] and [6], respectively. However, the presence of a bilinear stiffness in the cable dynamics (i.e., taut means hard spring and slack means soft spring), involves the combination of two Mathieu equations for correctly describing the cable-system dynamics.…”

Section: Introductionmentioning

confidence: 99%

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Experimental Demonstration of the Oscillatory Phenomenon Taut-Slack

Jordan¹,

Bustamante²

2006

Proceedings of the 45th IEEE Conference on Decision and Control

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In this paper a mechanism for reproducing the phenomenon known as "taut-slack" in a cable-body system is presented. The operation of the mechanism consists in forcing an oscillation of a point from which a body immersed in liquid is suspended by means of a cord. For particular values of ad-hoc design parameters of the mechanism, the body can oscillate nonlinearly mainly because the cable transits from a taut to a slack condition cyclically. Both body and suspension point positions are sensed in order to detect the periodicity of the resulting body motion. The modeling of the system dynamics yields a set of two Mathieu differential equations that are combined piecewise according to the cable taut-slack condition. The periodicity detection implemented in this work is based on the analysis of Cauchy series obtained from the Poincare maps. From the experimental results it could be obtained period doubling with the presence of the taut-slack phenomenon. Through comparison with simulated bifurcation diagrams and Poincare maps it could be noticed a good concordance between the signal profiles and also a comparable evidence of the behavior diversity.

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Section: Model Of the Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Experimental Demonstration of the Oscillatory Phenomenon Taut-Slack

Jordan¹,

Bustamante²

2006

Proceedings of the 45th IEEE Conference on Decision and Control

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“…Ramani et al (2004) presented an approach in which, by considering the quadratically damping term, the Strutt diagram shows a second bifurcation, see Figure 5.34, determined by the red line. This line is evaluated using a local perturbation method and the bifurcation line is better defined when closer to the point P. The point P is determined in the critical point where the derivative change its signal in the Strutt diagram first stability tongue.…”

Section: Parametric Resonance and Mathieu Instabilitymentioning

confidence: 99%

“…This line is evaluated using a local perturbation method and the bifurcation line is better defined when closer to the point P. The point P is determined in the critical point where the derivative change its signal in the Strutt diagram first stability tongue. The modal oscillator presented in Ramani et al (2004) …”

Section: Parametric Resonance and Mathieu Instabilitymentioning

confidence: 99%

“…Exploring the modal Mathieu-Hill oscillator, the experiment in water presents a non-linear term due to hydrodynamical damping mainly caused by drag forces, being necessary to use a perturbation techniques proposed by Ramani et al (2004), yielding to a local solution in the instability area in Strutt diagram. Finally, it is possible to study the modal stability by means of Strutt diagram, looking for results that might be inside any instability tongue.…”

Section: Introductionmentioning

confidence: 99%

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Experimental analysis of fluid-structure interaction phenomena on a vertical flexible cylinder: modal coeficients and parametric resonance.

Salles¹

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Partially extended oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping

Ishibashi

2024

Math Methods in App Sciences

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This study addressed the oscillation problems of half‐linear differential equations with periodic damping. The solution space of any linear equation is homogeneous and additive. Generally, by contrast, the solution space of half‐linear differential equations is homogeneous but not additive. Numerous oscillation and nonoscillation theorems have been devised for half‐linear differential equations featuring periodic functions as coefficients. However, in certain cases, such as applying Mathieu‐type differential equations to control engineering, which is a typical example of the Hill equation, some oscillation theorems cannot be applied. In this study, we established oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping. To prove the results, we used the Riccati technique and the composite function method, which focuses on the composite function of the indefinite integral of the coefficients of the target equation and an appropriate multivalued continuously differentiable function. Furthermore, we discuss the special case of the oscillation constant of a damped half‐linear Mathieu equation.

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Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation

Cited by 35 publications

References 2 publications

Experimental Demonstration of the Oscillatory Phenomenon Taut-Slack

Experimental Demonstration of the Oscillatory Phenomenon Taut-Slack

Experimental analysis of fluid-structure interaction phenomena on a vertical flexible cylinder: modal coeficients and parametric resonance.

Partially extended oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping

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