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Research Papers

On the Dynamics and Multiple Equilibria of an Inverted Flexible Pendulum With Tip Mass on a Cart

[+] Author and Article Information
Ojas Patil

Suman Mashruwala Advanced
Micro-Engineering Laboratory,
Department of Mechanical Engineering,
Indian Institute of Technology,
Bombay, Powai,
Mumbai 400076, India

Prasanna Gandhi

Suman Mashruwala Advanced
Micro-Engineering Laboratory,
Department of Mechanical Engineering,
Indian Institute of Technology,
Bombay, Powai,
Mumbai 400076, India
e-mail: gandhi@me.iitb.ac.in

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received June 22, 2012; final manuscript received February 9, 2014; published online April 11, 2014. Assoc. Editor: Qingze Zou.

J. Dyn. Sys., Meas., Control 136(4), 041017 (Apr 11, 2014) (9 pages) Paper No: DS-12-1195; doi: 10.1115/1.4026831 History: Received June 22, 2012; Revised February 09, 2014

Flexible link systems are increasingly becoming popular for advantages like superior performance in micro/nanopositioning, less weight, compact design, lower power requirements, and so on. The dynamics of distributed and lumped parameter flexible link systems, especially those in vertical planes are difficult to capture with ordinary differential equations (ODEs) and pose a challenge to control. A representative case, an inverted flexible pendulum with tip mass on a cart system, is considered in this paper. A dynamic model for this system from a control perspective is developed using an Euler Lagrange formulation. The major difference between the proposed method and several previous attempts is the use of length constraint, large deformations, and tip mass considered together. The proposed dynamic equations are demonstrated to display an odd number of multiple equilibria based on nondimensional quantity dependent on tip mass. Furthermore, the equilibrium solutions thus obtained are shown to compare fairly with static solutions obtained using elastica theory. The system is demonstrated to exhibit chaotic behavior similar to that previously observed for vibrating elastic beam without tip mass. Finally, the dynamic model is validated with experimental data for a couple of cases of beam excitation.

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Figures

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Fig. 1

Inverted flexible pendulum system with base excitation

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Fig. 2

Cantilevered flexible elastica column

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Fig. 3

Comparison of elastica solution and dynamic equilibrium solution for L = 0.3 m, w = 2 cm, and t = 0.4 mm

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Fig. 4

Variation of equilibrium tip displacement with load ratio P/Pcr

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Fig. 5

Experimental setup

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Fig. 6

Phase portraits: ωz = 5 rad/s, (ωz/ωn = 0.869), amplitudez = 2 cm

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Fig. 7

Phase portraits: ωz = 5 rad/s, (ωz/ωn = 0.869), amplitudez = 2.68 cm

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Fig. 8

Chaotic vibrations: ωz = 4 rad/s

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Fig. 9

Poincaré maps for frequency = 4 rad/s amplitudez = 1.45 cm: simulation and experimental results (bottom)

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