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Research Papers

The Udwadia–Kalaba Trajectory Control Applied to a Cantilever Beam—Experimental Results

[+] Author and Article Information
Raphael Pereira Spada

Department of Mechanical Engineering,
São Carlos School of Engineering,
University of São Paulo,
Trabalhador São-Carlense 400,
São Carlos 13566-590, Brazil
e-mail: spadaeu@yahoo.com.br

Rodrigo Nicoletti

Department of Mechanical Engineering,
São Carlos School of Engineering,
University of São Paulo,
Trabalhador São-Carlense 400,
São Carlos 13566-590, Brazil
e-mail: rnicolet@sc.usp.br

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 2, 2015; final manuscript received March 3, 2017; published online June 5, 2017. Assoc. Editor: Tarunraj Singh.

J. Dyn. Sys., Meas., Control 139(9), 091002 (Jun 05, 2017) (6 pages) Paper No: DS-15-1606; doi: 10.1115/1.4036236 History: Received December 02, 2015; Revised March 03, 2017

The Udwadia–Kalaba methodology is a possible way of explicitly obtaining the equations of motion of constrained systems. From these equations of motion, one can estimate the necessary forces in the constraint to keep the system in a given motion. Hence, the Udwadia–Kalaba methodology can also apply to active tracking control of subsystems or the control of points of a structure. In this work, one investigates experimentally the benefits and drawbacks of such control strategy by applying it to the control of out-of-plane vibrations of a cantilever beam. The beam is excited by a shaker mounted near the clamped end of the beam. A second shaker applies the control forces in the free end of the beam, where an accelerometer is used for feedback. The vibration behavior of the beam under excitation/control is measured by a laser vibrometer. Results show that the methodology changes the dynamic behavior of the structure by changing its boundary conditions at the point of control, thus shifting natural frequencies and mode shapes. Results also show that the successful implementation of the method experimentally is sensitive to the quality of modeling of the structure.

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Figures

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Fig. 1

Experimental setup: (1) cantilever beam, (2) excitation shaker (with load cell), (3) control shaker, (4) laser vibrometer, and (5) accelerometer (control loop feedback)

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Fig. 2

Beam under study: measuring points, excitation point, and control point

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Fig. 3

Vibration modes of the system in the frequency range of 2–350 Hz (comparison between the updated model and the experiment): (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, and (e) fifth mode

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Fig. 4

Frequency response function of the shaker + amplifier system used as control actuator (experiment)

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Fig. 5

Block diagram of the controller implemented experimentally (matlab simulink and NI PCI 6229 acquisition board)

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Fig. 6

First vibration mode of the beam: comparison between the controlled system (experiment) and the clamped-simply supported beam (model): (a) C1 = C2 = 2500 and (b) C1 = C2 = 5000

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Fig. 7

Second vibration mode of the beam: comparison between the controlled system (experiment) and the clamped-simply supported beam (model): (a) C1 = C2 = 2500 and (b) C1 = C2 = 5000

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Fig. 8

Third vibration mode of the beam: comparison between the controlled system (experiment) and the clamped-simply supported beam (model): (a) C1 = C2 = 2500 and (b) C1 = C2 = 5000

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Fig. 9

Fourth vibration mode of the beam: comparison between the controlled system (experiment) and the clamped-simply supported beam (model): (a) C1 = C2 = 2500 and (b) C1 = C2 = 5000

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