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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Zhiling, T., 2004, “Modeling and Control of Flexible Link Robots,” M.S. thesis, National University of Singapore, Singapore.
Deshmukh, S., and Gandhi, P., 2009, “Optomechanical Scanning Systems for Microstereolithography (MSL): Analysis and Experimental Verification,” J. Mater. Process. Technol., 209(3), pp. 1275–1285. [CrossRef]
Macchelli, A., and Melchiorri, C., 2004, “Modeling and Control of the Timoshenko Beam: The Distributed Port Hamiltonian Approach,” SIAM J. Control Optim., 43(2), pp. 743–767. [CrossRef]
Voß, T., Scherpen, J., and Onck, P., 2008, “Modeling for Control of an Inflatable Space Reflector, the Nonlinear 1-D Case,” Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, pp. 1777–1782.
Banavar, R., and Dey, B., 2010, “Stabilizing a Flexible Beam on a Cart: A Distributed Port-Hamiltonian Approach,” J. Nonlinear Sci., 20(2), pp. 131–151. [CrossRef]
Trivedi, M., Banavar, R., and Maschke, B., 2011, “Modeling of Hybrid Lumped-Distributed Parameter Mechanical Systems With Multiple Equilibria,” Proceedings of the 18th IFAC World Congress, Milano, Italy.
To, C., 1982, “Vibration of a Cantilever Beam With Base Excitation and a Tip Mass,” J. Sound Vib., 83(4), pp. 445–460. [CrossRef]
Esmailzadeh, E., and Nakhaie-Jazar, G., 1998, “Periodic Behaviour of a Cantilever Beam With End Mass Subjected to Harmonic Base Excitation,” Int. J. Nonlinear Mech., 33(4), pp. 566–577. [CrossRef]
Tang, J., and Ren, G., 2009, “Modeling and Simulation of a Flexible Inverted Pendulum System,” Tsinghua Sci. Technol., 14(S2), pp. 22–26. [CrossRef]
Xu, C., and Yu, X., 2004, “Mathematical Modeling of the Inverted Flexible Pendulum Control System,” J. Control Theory Appl., 83(3), pp. 281–282. [CrossRef]
Kong, K., 2009, “Fuzzy Logic PD Control of a Non-Linear Inverted Flexible Pendulum System,” M.S. thesis, California State University, Chico, CA.
Sheheitli, H., and Rand, R., 2012, “On the Dynamics of a Thin Elastica,” Int. J. Nonlinear Mech., 47, pp. 99–107. [CrossRef]
Pak, C. H., Rand, R. H., and Moon, F., 1992, “Free Vibrations of a Thin Elastica by Normal Modes,” Nonlinear Dyn., 3, pp. 347–364. [CrossRef]
Cusumano, J., and Moon, F., 1995, “Chaotic Non-Planar Vibrations of the Thin Elastica, Part I—Experimental Observation of Planar Instability,” J. Sound Vib., 179(2), pp. 185–208. [CrossRef]
Cusumano, J., and Moon, F., 1995, “Chaotic Non-Planar Vibrations of the Thin Elastica, Part II—Derivation and Analysis of a Low Dimensional Model,” J. Sound Vib., 179(2), pp. 209–226. [CrossRef]
Mann, B., 2009, “Energy Criterion for Potential Well Escapes in a Bistable Magnetic Pendulum,” J. Sound Vib., 323, pp. 864–876. [CrossRef]
Litak, G., and Coccolo, M., 2011, “Nonlinear Oscillations of an Elastic Inverted Pendulum,” Proceedings of the 4th IEEE International Conference on Nonlinear Science and Complexity, Budapest, Hungary, Aug. 6–11.
Meirovitch, L., 1975, Elements of Vibration Analysis, McGraw-Hill, New York.
Laura, P., Pombo, J., and Susemihl, E., 1974, “A Note on the Vibrations of a Clamped Free Beam With a Mass at the Free End,” J. Sound Vib., 37(2), pp. 161–168. [CrossRef]
Lanczos, C., 1952, The Variational Principles of Mechanics. Mathematical Expositions, No. 4, University of Toronto Press, Toronto.
Borg, S., 1961, Fundamentals of Engineering Elasticity, D. Van Nostrand Company, Princeton, NJ.
Gere, J., 2004, Mechanics of Materials, 5th ed., Thomson Asia, Singapore.
Lepik, U., 1995, “Elastic Plastic Vibrations of a Buckled Beam,” Int. J. Nonlinear Mech., 30(2), pp. 129–139. [CrossRef]


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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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