Transition curves in a parametrically excited pendulum with a force of elliptic type

Abstract

This article investigates the equilibria and stability of a pendulum when the support has a prescribed motion defined by an elliptic function. Stability charts are generated in the parameter plane for different values of the elliptic function modulus. Numerical integration and Floquet theory are used to generate stability charts that are later obtained through harmonic balance analysis. It is shown that the size and location of the instability tongues is directly linked to the elliptic function modulus. Comparisons are also made between the stability charts of Mathieu's equation and those of the pendulum when the prescribed motion is defined by an elliptic function.

Jacobi elliptic functions: A review of nonlinear oscillatory application problems

“…It illustrates clearly that as m increases from negative values approaching unity, the instability regions shift towards lower values of δ and become more dense when m approaches unity. This conclusion agrees with the results shown in [76] for 0 o m o1.…”

“…This is shown here for the Cases I and II from Section 2.2, based on the results from [76,77]. Transition curves emanate from the so-called zero-points, which represent resonant values of δ 0 ¼ δ ε ¼ 0 ð Þ.…”

“…Artim and Aydin [75] examined the influence of an axial force that is the sum of the squares of all three basic Jacobi elliptic functions. Sah and Mann [76] considered small vibrations of a planar undamped pendulum whose pivot point has a prescribed acceleration defined by the Jacobi cn function. Sah and Mann obtained the corresponding stability chart with transition curves and resonance tongues for the elliptic parameter ranging from zero to unity.…”

Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

“…Applied Mechanics Reviews. Received August 07, 2017; Accepted manuscript posted January 31, 2018. doi:10.1115/1.4039144 Copyright (c) 2018 by ASME cases can be distinguished [53] (indicated subsequently by the corresponding index): Case I corresponds to m I < 0; Case II includes the domain 0 < m II < 1 (this case was also analysed in [54]); Case III encompasses m III > 1. Note that the value m = 1 is not of interest here as the cn elliptic function turns then into the hyperbolic secant (the inverse of hyperbolic cosine), which is not periodic.…”

Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator

“…where cn( , 2 ) is the Jacobian elliptic function that has a period in equal to 4 ( 2 ), and ( 2 ) is the complete elliptic integral of the first kind for the modulus [16,17]. Note that 0 ≤ | 2 | < 1 must hold, and when the modulus of cn( , 2 ) is zero, then cn( , 2 ) and the trigonometric function cos( ) coincide and, thus, (2) reduces to (1).…”