Computing the stability regions of Hill's equation

Simakhina, Svetlana V.; Tier, Charles

doi:10.1016/j.amc.2004.01.002

Cited by 13 publications

(7 citation statements)

References 5 publications

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“…Note that one particular case of the Hill's equation is the Mathieu equation, where f ðtÞ ¼ cosðtÞ. Methods to find approximate solutions and construct stability maps for the Mathieu equation have been extensively applied, 14,18,19 and served partially as inspiration for the procedure developed here. Invoking the nondimensional time ¼ !…”

Section: Brief Discussion On the Selected Methodsmentioning

confidence: 99%

Theoretical investigation of the particle response to an acoustic field

Achury

Polifke

2016

International Journal of Spray and Combustion Dynamics

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In this paper, the problem of a particle subjected to an acoustic field is addressed theoretically. Once the fundamental equation of motion is obtained, two nonlinearities are identified: one related to the drag law and one associated with the excitation. In order to face the nonlinearities, two cases are constructed: the first corresponds to the parametric numerical solution of a particle with nonlinear drag in an oscillating flow field (infinite wavelength) and the second refers to the particle submitted to an acoustic standing wave (finite wavelength). For the latter, an approximated analytical solution is formulated. The system is linearized around an equilibrium point and the parameters of the equation are grouped in three nondimensional numbers: the Stokes number (S t), the acoustic Mach number (M a), and the densities ratio (). Conditions of parametric resonance in the particle response are deduced for this system by means of the analytical method here proposed, based on Hill's determinants. Comparison with numerical solutions of the linearized and nonlinearized equations close to an equilibrium point corroborates the analysis for different combinations of parameters.

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Section: Brief Discussion On the Selected Methodsmentioning

confidence: 99%

Theoretical investigation of the particle response to an acoustic field

Achury

Polifke

2016

International Journal of Spray and Combustion Dynamics

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“…Parametric excitation by V 1 cos(Ωt) causes n − 1 tongues of instability emanating from the horizontal axis V 1 = 0 which get wider as V 1 increases. The tongues of instability, aka Arnold tongues [24], arise for periodic systems, including the Mathieu equation, and their properties have been studied extensively for the more general case of Hill equations [25,26,27]. For n 10 the stability regions show universal behaviour: a quadratic increase up to V n /2V A 0.7 followed by a linear decrease to zero at V n /2V A = 0.91.…”

Section: Parametric Resonancementioning

confidence: 99%

Two-frequency operation of a Paul trap to optimise confinement of two species of ions

Foot

Trypogeorgos

Bentine

et al. 2018

International Journal of Mass Spectrometry

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We describe the operation of an electrodynamic ion trap in which the electric quadrupole field oscillates at two frequencies. This mode of operation allows simultaneous tight confinement of ions with extremely different charge-to-mass ratios, e.g., singly ionised atomic ions together with multiply charged nanoparticles. We derive the stability conditions for two-frequency operation from asymptotic properties of the solutions of the Mathieu equation and give a general treatment of the effect of damping on parametric resonances. Two-frequency operation is effective when the two species' mass ratios and charge ratios are sufficiently large, and further when the frequencies required to optimally trap each species are widely separated. This system resembles two coincident Paul traps, each operating close to a frequency optimized for one of the species, such that both species are tightly confined. This method of operation provides an advantage over single-frequency Paul traps, in which the more weakly confined species forms a sheath around a central core of tightly confined ions. We verify these ideas using numerical simulations and by measuring the parametric heating induced in experiments by the additional driving frequency.

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“…The most common means of determining the stability of Mathieu's equation is using a Strutt diagram [20]. A Strutt diagram is a plot of the parameters d versus .…”

Section: ̈( ) [ ( )] ( ) [mentioning

confidence: 99%

Analytical model and stability analysis of the leading edge spar of a passively morphing ornithopter wing

Wissa¹,

Calogero²,

Wereley³

et al. 2015

Bioinspir. Biomim.

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This paper presents the stability analysis of the leading edge spar of a flapping wing unmanned air vehicle with a compliant spine inserted in it. The compliant spine is a mechanism that was designed to be flexible during the upstroke and stiff during the downstroke. Inserting a variable stiffness mechanism into the leading edge spar affects its structural stability. The model for the spar-spine system was formulated in terms of the well-known Mathieu's equation, in which the compliant spine was modeled as a torsional spring with a sinusoidal stiffness function. Experimental data was used to validate the model and results show agreement within 11%. The structural stability of the leading edge spar-spine system was determined analytically and graphically using a phase plane plot and Strutt diagrams. Lastly, a torsional viscous damper was added to the leading edge spar-spine model to investigate the effect of damping on stability. Results show that for the un-damped case, the leading edge spar-spine response was stable and bounded; however, there were areas of instability that appear for a range of spine upstroke and downstroke stiffnesses. Results also show that there exist a damping ratio between 0.2 and 0.5, for which the leading edge spar-spine system was stable for all values of spine upstroke and downstroke stiffnesses.

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Computing the stability regions of Hill's equation

Cited by 13 publications

References 5 publications

Theoretical investigation of the particle response to an acoustic field

Theoretical investigation of the particle response to an acoustic field

Two-frequency operation of a Paul trap to optimise confinement of two species of ions

Analytical model and stability analysis of the leading edge spar of a passively morphing ornithopter wing

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