Research Papers

Numerical Location of Painlevé Paradox-Associated Jam and Lift-Off in a Double-Pendulum Mechanism

[+] Author and Article Information
Shane J. Burns

Institute for Infrastructure and Environment,
School of Engineering,
The University of Edinburgh,
Edinburgh EH9 3JL, UK
e-mail: shane.burns111@gmail.com

Petri T. Piiroinen

School of Mathematics,
Statistics and Applied Mathematics,
National University of Ireland, Galway,
University Road,
Galway H91 TK33, Ireland
e-mail: petri.piiroinen@nuigalway.ie

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 2, 2016; final manuscript received June 6, 2017; published online September 7, 2017. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(6), 061007 (Sep 07, 2017) (8 pages) Paper No: CND-16-1530; doi: 10.1115/1.4037033 History: Received November 02, 2016; Revised June 06, 2017

In this article, we will introduce the phenomenon known as the Painlevé paradox and further discuss the associated coupled phenomena, jam and lift-off. We analyze under what conditions the Painlevé paradox can occur for a general two-body collision using a framework that can be easily used with a variety of impact laws, however, in order to visualize jam and lift-off in a numerical simulation, we choose to use a recently developed energetic impact law as it is capable of achieving a unique forward solution in time. Further, we will use this framework to derive the criteria under which the Painlevé paradox can occur in a forced double-pendulum mechanical system. First, using a graphical technique, we will show that it is possible to achieve the Painlevé paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlevé paradox in the double-pendulum system.

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Grahic Jump Location
Fig. 1

(a) Geometry of two planar rigid bodies H and H′ in contact at a point C. (b) A free-body force diagram corresponding to (a). Adapted from [21], copyright 2015 Elsevier LTD. Adapted with permission from the authors.

Grahic Jump Location
Fig. 2

(a) A forced double pendulum in which the lower mass m2 can impact with a noncompliant vertical plane. The distance between the upper joint and the plane is denoted by d. (b) The forced double pendulum in constrained contact with the plane moving vertically with velocity V. The belt consists of two regions, one with the lower coefficient of friction value μ, one with the higher value μp.

Grahic Jump Location
Fig. 3

The terms B and C as functions of ϕ for θ = 4.2062, l1 = 0.2, l2 = 0.25, and m1 = m2 = 0.5. In (a), B and C are plotted for 0 ≤ θ ≤ 2π, and in (b) a magnification of region I in (a) is shown.

Grahic Jump Location
Fig. 5

Overall dynamics before and after lift-off for θ = 4.3, ϕ=4.2062, V = −0.98, e = 0.9, and μ = 0.3. (a) A plot of the normal contact point velocity c˙2 as a function of time for μp: = μa = 0.461365 and μp: = μb = 0.461366. (b) Time histories for the generalized coordinates θ and ϕ for μp: = μb = 0.461366. (c) Time histories for the angular velocities θ˙ and ϕ˙ for μp: = μb = 0.461366.

Grahic Jump Location
Fig. 4

Four regions in the space θ,ϕ∈[0,2π) for which the Painlevé paradox can occur for a friction coefficient μ < 1. In (a), all regions are shown without any constraints, and in (b) the constraint that d must be negative is imposed such that the shaded area represents the regions in which d is positive, and initial conditions in this region can thus not be selected.



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