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Research Papers

Dynamics of a Self-Balancing Double-Pendulum System

[+] Author and Article Information
Krystof P. Jankowski

Magna Closures of America, Inc.,
39600 Lewis Drive,
Novi, MI 48377
e-mail: krystof.jankowski@magna.com

Andrzej Mitura

Department of Applied Mechanics,
Lublin University of Technology,
Nadbystrzycka 36,
Lublin 20-618, Poland
e-mail: a.mitura@pollub.pl

Jerzy Warminski

Department of Applied Mechanics,
Lublin University of Technology,
Nadbystrzycka 36,
Lublin 20-618, Poland
e-mail: j.warminski@pollub.pl

1Corresponding author.

Manuscript received February 2, 2015; final manuscript received October 13, 2015; published online December 11, 2015. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 11(4), 041012 (Dec 11, 2015) (9 pages) Paper No: CND-15-1035; doi: 10.1115/1.4031978 History: Received February 02, 2015; Revised October 13, 2015

Modeling and analysis of a system of two self-balancing pendulums is presented in this paper. Such systems are commonly used as elements of automotive door latch mechanisms that can be subjected to oscillatory excitation or vibratory inertia forces occurring during crash events. In order to avoid an unwanted behavior such as opening of the door, the considered mechanism should be properly designed and its dynamical response well understood and predictable. One pendulum of the double-pendulum system, playing the role of a counterweight (CW), is used to reduce the second (or main) pendulum motion under inertia loading. The interaction force between the pendulums is defined as the reaction of a holonomic constraint linking the rotations of both pendulums. Another reaction force acts between one of the pendulums and the support, reinforced by the action of a preloaded spring. An important aspect of the model is its discontinuous nature due to the presence of a gap in the interface area. This may result in impacts between both pendulums and between one of the pendulums and the support. High-frequency/high-acceleration amplitude vibratory motion of the base part provides inertia input to the system. Classical multibody dynamics approach is adopted first to solve the equations of motion. It is shown that the considered system under certain conditions responds with a high-amplitude irregular motion. A special methodology is used in order to study the regions of chaotic motion, with the goal to gain more understanding of the considered system dynamics. Bifurcation diagrams are presented together with quantitative and qualitative analysis of the motion. The sensitivity of solutions to variation of system parameters and input characteristics is also analyzed in the paper.

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References

Figures

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Fig. 1

Solid-body representation of a double-pendulum system

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Fig. 2

Double-pendulum system on a vibrating support: (a) side view (X–Y plane) and (b) front view (Y–Z plane)—views not to scale

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Fig. 3

Directions of interaction forces during contact on the lower buffer

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Fig. 4

Interaction forces and moment arms during contact on the lower and upper buffers

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Fig. 5

CW mass needed to balance the main pendulum versus pendulum rotation angle

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Fig. 6

Double-pendulum system responses to an exemplary acceleration input (h = 0.003 m)

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Fig. 7

Double-pendulum system responses to an exemplary acceleration input (h = 0.001 m)

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Fig. 8

Responses to a harmonic acceleration input with ω = 1257 rad/s, A = 400 G's, and h = 0.003 m

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Fig. 9

Bifurcation diagram: the main pendulum rotation angle ϕ for ω ∈ [10,500] rad/s, A = 100 G's, and h = 0.003 m

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Fig. 10

Bifurcation diagram: the main pendulum rotation angle ϕ for ω ∈ [10, 3000] rad/s, A = 100 G's: (a) h = 0 m and (b) h = 0.0001 m

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Fig. 11

Poincaré's maps for ω = 4000 rad/s, A = 50 G's: (a) h = 0 m and (b) h = 0.003 m

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Fig. 12

Plots of the quality index: (a) J(A, ω): h = 0 m and (b) J(h, ω): A = 50 G's

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Fig. 13

Quality index variation with the change of the CW parameters (h = 0 m, A = 50 G's): (a) J(S1, ω): S2 = 1 and (b) J(S2, ω): S1 = 1

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Fig. 14

Quality index variation with the change of the main pendulum parameters (h = 0 m, A = 50 G's): (a) J(S3, ω): S4 = 1 and (b) J(S4, ω): S3 = 1

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Fig. 15

Bifurcation diagram for the main pendulum rotation angle with h = 0 m and A = 50 G's

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Fig. 16

Time histories with h = 0 m and A = 50 G's: (a) ω = 1487.945 rad/s and (b) ω = 1487.95 rad/s

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Fig. 17

Transition from one form of vibrations to another (h = 0 m, A = 50 G's): ω = 1487.945 rad/s

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