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Technical Brief

Further Insights Into the Damping-Induced Self-Recovery Phenomenon

[+] Author and Article Information
Tejas Kotwal

Department of Mathematics,
Indian Institute of Technology Bombay,
Mumbai, India
e-mail: tejas.kotwal@iitb.ac.in

Roshail Gerard

Department of Mechanical Engineering,
Indian Institute of Technology Bombay,
Mumbai, India

Ravi Banavar

Systems and Control Engineering,
Indian Institute of Technology Bombay,
Mumbai, India

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received January 4, 2018; final manuscript received November 28, 2018; published online January 14, 2019. Assoc. Editor: Heikki Handroos.

J. Dyn. Sys., Meas., Control 141(4), 044504 (Jan 14, 2019) (3 pages) Paper No: DS-18-1007; doi: 10.1115/1.4042144 History: Received January 04, 2018; Revised November 28, 2018

In a series of papers, Chang et al. proved and experimentally demonstrated a phenomenon in underactuated mechanical systems, that they termed “damping-induced self-recovery.” This paper further investigates a few features observed in these demonstrated experiments and provides additional theoretical interpretation for the same. In particular, we present a model for the infinite-dimensional fluid–stool–wheel system, that approximates its dynamics to that of the better understood finite dimensional case, and comment on the effect of the intervening fluid on the large amplitude oscillations observed in the bicycle wheel–stool experiment.

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References

Ruina, A. , 2010, “ Dynamic Walking 2010: Cats, Astronauts, Trucks, Bikes, Arrows, and Muscle-Smarts: Stability, Translation, and Rotation,” Massachusetts Institute of Technology, Cambridge, MA, Dec. 11, 2018, http://robots.ihmc.us/dynamicwalking2018/
Chang, D. E. , and Jeon, S. , 2013, “ Damping-Induced Self Recovery Phenomenon in Mechanical Systems With an Unactuated Cyclic Variable,” ASME J. Dyn. Syst. Meas. Control, 135(2), p. 021011. [CrossRef]
Chang, D. E. , and Jeon, S. , 2013, “ On the Damping-Induced Self-Recovery Phenomenon in Mechanical Systems With Several Unactuated Cyclic Variables,” J. Nonlinear Sci., 23(6), pp. 1023–1038. [CrossRef]
Chang, D. E. , and Jeon, S. , 2013, “ On the Self-Recovery Phenomenon in the Process of Diffusion,” eprint arXiv:1305.6658. https://arxiv.org/abs/1305.6658
Chang, D. E. , and Jeon, S. , 2015, “ On the Self-Recovery Phenomenon for a Cylindrical Rigid Body Rotating in an Incompressible Viscous Fluid,” ASME J. Dyn. Syst. Meas. Control, 137(2), p. 021005. [CrossRef]
Landau, L. D. , and Lifshitz, E. M. , 1959, Fluid Mechanics, Vol. 6, Elsevier, Amsterdam, The Netherlands.
Farlow, S. J. , 1993, Partial Differential Equations for Scientists and Engineers, Courier Corporation, Chelmsford, MA.

Figures

Grahic Jump Location
Fig. 1

A schematic diagram of the wheel–stool model

Grahic Jump Location
Fig. 2

Trajectory of the wheel with control parameters c0=1 and c1=3 (solid blue line; this is a case of overshoot alone) and c0=1 and c1=1 (dotted blue line; this is a case of oscillations). The red dashed line denotes the desired trajectory, and the overshoot of the wheel may be seen at t=2s.

Grahic Jump Location
Fig. 3

Trajectory of the stool under the influence of the wheel trajectory shown in Fig. 2. The black dashed line denotes the zero position of the stool, and the overshoot/oscillations of the stool may be seen after it recovers (Note the slight delay when compared to the overshoot of the wheel; this is due to the time constant of the system).

Grahic Jump Location
Fig. 4

Trajectory of the stool under the influence of the wheel trajectory shown in Fig. 2 with control parameters c0=1 and c1=1 and different damping functions k(ϕs). The black dashed line denotes the zero position of the stool, while the solid blue line denotes the position of the stool when k(ϕs)=k/2π, dashed blue line when k(ϕs)=k(1+cos(ϕs))/2π, and dotted blue line when k(ϕs)=2k(cos(ϕs))2/π. The constants chosen are k =1, Iw=0.0625, and Is=0.625 in model (3).

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