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Research Papers

Lyapunov-Based Boundary Control of Strain Gradient Microscale Beams With Exponential Decay Rate

[+] Author and Article Information
Ramin Vatankhah

School of Mechanical Engineering,
Shiraz University,
Shiraz, Iran
e-mail: rvatankhah@shirazu.ac.ir

Ali Najafi, Hassan Salarieh, Aria Alasty

Center of Excellence in Design, Robotics,
and Automation (CEDRA),
School of Mechanical Engineering,
Sharif University of Technology,
Tehran, Iran

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 25, 2013; final manuscript received October 27, 2014; published online January 27, 2015. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 137(3), 031003 (Jun 01, 2015) (12 pages) Paper No: VIB-13-1132; doi: 10.1115/1.4028964 History: Received April 25, 2013; Revised October 27, 2014; Online January 27, 2015

In nonclassical microbeams, the governing partial differential equation (PDE) of the system and corresponding boundary conditions are obtained based on the nonclassical continuum mechanics. In this study, exponential decay rate of a vibrating nonclassical microscale Euler–Bernoulli beam is investigated using a linear boundary control law and by implementing a proper Lyapunov functional. To illustrate the performance of the designed controllers, the closed-loop PDE model of the system is simulated via finite element method (FEM). To this end, new nonclassical beam element stiffness and mass matrices are developed based on the strain gradient theory and verification of this new beam element is accomplished in this work.

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Figures

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Fig. 1

Normalized variables in the first simulation: (a) deflection, (b) slope, (c) velocity, and (d) angular velocity; of the tip of the stabilized beam versus normalized time

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Fig. 2

Normalized tip force as the control effort versus normalized time in the first simulation

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Fig. 3

Behavior of the strain gradient Euler–Bernoulli microbeam in the first simulation: (a) open-loop response and (b) closed-loop response

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Fig. 4

Variation of the function V(t)-V(0)exp(-ct) versus normalized time in the first simulation

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Fig. 5

Normalized variables in the second simulation: (a) deflection, (b) slope, (c) velocity, and (d) angular velocity; of the tip of the stabilized beam versus normalized time

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Fig. 6

Normalized tip force as the control effort versus normalized time in the second simulation

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Fig. 7

Behavior of the closed-loop system in the second simulation

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