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.

References

1.
Lun
,
F. Y.
,
Zhang
,
P.
,
Gao
,
F. B.
, and
Jia
,
H. G.
,
2006
, “
Design and Fabrication of Micro-Optomechanical Vibration Sensor
,”
Microfabr. Technol.
,
120
(
1
), pp.
61
64
.
2.
Moser
,
Y.
, and
Gijs
,
M. A. M.
,
2007
, “
Miniaturized Flexible Temperature Sensor
,”
J. Microelectromech. Syst.
,
16
(
6
), pp.
1349
1354
.10.1109/JMEMS.2007.908437
3.
De Boer
,
M. P.
,
Luck
,
D. L.
,
Ashurst
,
W. R.
,
Maboudian
,
R.
,
Corwin
,
A. D.
,
Walraven
,
J. A.
, and
Redmond
,
J. M.
,
2004
, “
High-Performance Surface-Micromachined Inchworm Actuator
,”
J. Microelectromech. Syst.
,
13
(
1
), pp.
63
74
.10.1109/JMEMS.2003.823236
4.
Batra
,
R. C.
,
Porfiri
,
M.
, and
Spinello
,
D.
,
2008
, “
Vibrations of Narrow Microbeams Predeformed by an Electric Field
,”
J. Sound Vib.
,
309
(
3–5
), pp.
600
612
.10.1016/j.jsv.2007.07.030
5.
Fleck
,
N. A.
,
Muller
,
G. M.
,
Ashby
,
M. F.
, and
Hutchinson
,
J. W.
,
1994
, “
Strain Gradient Plasticity: Theory and Experiment
,”
Acta Metall. Mater.
,
42
(
2
), pp.
475
487
.10.1016/0956-7151(94)90502-9
6.
Ma
,
Q.
, and
Clarke
,
D. R.
,
1995
, “
Size Dependent Hardness of Silver Single Crystals
,”
J. Mater. Res.
,
10
(
4
), pp.
853
863
.10.1557/JMR.1995.0853
7.
Stolken
,
J. S.
, and
Evans
,
A. G.
,
1998
, “
A Microbend Test Method for Measuring the Plasticity Length Scale
,”
Acta Mater.
,
46
(
14
), pp.
5109
5115
.10.1016/S1359-6454(98)00153-0
8.
Lam
,
D. C. C.
,
Yang
,
F.
,
Chong
,
A. C. M.
,
Wang
,
J.
, and
Tong
,
P.
,
2003
, “
Experiments and Theory in Strain Gradient Elasticity
,”
J. Mech. Phys. Solids
,
51
(
8
), pp.
1477
1508
.10.1016/S0022-5096(03)00053-X
9.
Mindlin
,
R. D.
, and
Tiersten
,
H. F.
,
1962
, “
Effects of Couple-Stresses in Linear Elasticity
,”
Arch. Ration. Mech. Anal.
,
11
(
1
), pp.
415
448
.10.1007/BF00253946
10.
Toupin
,
R. A.
,
1962
, “
Elastic Materials With Couple-Stresses
,”
Arch. Ration. Mech. Anal.
,
11
(
1
), pp.
385
414
.10.1007/BF00253945
11.
Koiter
,
W. T.
,
1964
, “
Couple Stresses in the Theory of Elasticity, I and II
,” Nederl. Akad. Wetensch. Proc. Ser. B,
67
, pp.
17
29
.
12.
Yang
,
F.
,
Chong
,
A. C. M.
,
Lam
,
D. C. C.
, and
Tong
,
P.
,
2002
, “
Couple Stress Based Strain Gradient Theory for Elasticity
,”
Int. J. Solids Struct.
,
39
(
10
), pp.
2731
2743
.10.1016/S0020-7683(02)00152-X
13.
Park
,
S. K.
, and
Gao
,
X. L.
,
2006
, “
Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory
,”
J. Micromech. Microeng.
,
16
(
11
), pp.
2355
2359
.10.1088/0960-1317/16/11/015
14.
Kong
,
S.
,
Zhou
,
S.
,
Nie
,
Z.
, and
Wang
,
K.
,
2008
, “
The Size-Dependent Natural Frequency of Bernoulli–Euler Micro-Beams
,”
Int. J. Eng. Sci.
,
46
(
5
), pp.
427
437
.10.1016/j.ijengsci.2007.10.002
15.
Ma
,
H. M.
,
Gao
,
X. L.
, and
Reddy
,
J. N.
,
2008
, “
A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory
,”
J. Mech. Phys. Solids
,
56
(
12
), pp.
3379
3391
.10.1016/j.jmps.2008.09.007
16.
Kong
,
S.
,
Zhou
,
S.
,
Nie
,
Z.
, and
Wang
,
K.
,
2009
, “
Static and Dynamic Analysis of Micro Beams Based on Strain Gradient Elasticity Theory
,”
Int. J. Eng. Sci.
,
47
(
4
), pp.
487
498
.10.1016/j.ijengsci.2008.08.008
17.
Zhao
,
J.
,
Zhou
,
S.
,
Wang
,
B.
, and
Wang
,
X.
,
2012
, “
Nonlinear Microbeam Model Based on Strain Gradient Theory
,”
Appl. Math. Modell.
,
36
(
6
), pp.
2674
2686
.10.1016/j.apm.2011.09.051
18.
Asghari
,
M.
, and
Taati
,
E.
,
2013
, “
A Size-Dependent Model for Functionally Graded Micro-Plates for Mechanical Analyses
,”
J. Vib. Control
,
19
(11), pp. 1614–1632.10.1177/1077546312442563
19.
Vatankhah
,
R.
,
Kahrobaiyan
,
M. H.
,
Alasty
,
A.
, and
Ahmadian
,
M. T.
,
2013
, “
Nonlinear Forced Vibration of Strain Gradient Microbeams
,”
Appl. Math. Modell.
,
37
(
18
), pp.
8363
8382
.10.1016/j.apm.2013.03.046
20.
Zhang
,
W.
,
Meng
,
G.
, and
Li
,
H.
,
2006
, “
Adaptive Vibration Control of Micro-Cantilever Beam With Piezoelectric Actuator in MEMS
,”
Int. J. Adv. Manuf. Technol.
,
28
(
3
), pp.
321
327
.10.1007/s00170-004-2363-5
21.
Yen
,
J. Y.
,
Lan
,
K. J.
, and
Kramar
,
J. A.
,
2005
, “
Active Vibration Isolation of a Large Stroke Scanning Probe Microscope by Using Discrete Sliding Mode Control
,”
Sens. Actuators A
,
121
(
1
), pp.
243
250
.10.1016/j.sna.2005.02.035
22.
Wang
,
P. K. C.
,
1998
, “
Feedback Control of Vibrations in a Micromachined Cantilever Beam With Electrostatic Actuators
,”
J. Sound Vib.
,
213
(
3
), pp.
537
550
.10.1006/jsvi.1998.1525
23.
Yu
,
Y.
,
Zhang
,
X. N.
, and
Xie
,
S. L.
,
2009
, “
Optimal Shape Control of a Beam Using Piezoelectric Actuators With Low Control Voltage
,”
Smart Mater. Struct.
,
18
(
9
), p.
095006
.10.1088/0964-1726/18/9/095006
24.
Kharrat
,
C.
,
Colinet
,
E.
, and
Voda
,
A.
,
2008
, “
A Robust Control Method for Electrostatic Microbeam Dynamic Shaping With Capacitive Detection
,” 17th International Federation of Automatic Control World Congress (IFAC 2008), Seoul, South Korea, July 6–11, pp.
568
573
.
25.
Vatankhah
,
R.
,
Najafi
,
A.
,
Salarieh
,
H.
, and
Alasty
,
A.
,
2013
, “
Boundary Stabilization of Non-Classical Micro-Scale Beams
,”
Appl. Math. Modell.
,
37
(
20
), pp.
8709
8724
.10.1016/j.apm.2013.03.048
26.
Vatankhah
,
R.
,
Najafi
,
A.
,
Salarieh
,
H.
, and
Alasty
,
A.
,
2014
, “
Exact Boundary Controllability of Vibrating Non-Classical Euler–Bernoulli Micro-Scale Beams
,”
J. Math. Anal. Appl.
,
418
(
2
), pp.
985
997
.10.1016/j.jmaa.2014.03.012
27.
Najafi
,
A.
,
Eghtesad
,
M.
,
Daneshmand
,
F.
, and
Lotfazar
,
A.
,
2011
, “
Boundary Stabilization of Parachute Dams in Contact With Fluid
,”
ASME J. Vib. Acoust.
,
133
(
6
), p.
061009
.10.1115/1.4004662
28.
Canbolat
,
H.
,
Dawson
,
D.
,
Rahn
,
C.
, and
Vedagarbha
,
P.
,
1998
, “
Boundary Control of a Cantilevered Flexible Beam With Point-Mass Dynamics at the Free End
,”
Mechatronics
,
8
(
2
), pp.
163
186
.10.1016/S0957-4158(97)00022-6
29.
Fard
,
M. P.
, and
Sagatun
,
S. I.
,
2001
, “
Exponential Stabilization of a Transversely Vibrating Beam Via Boundary Control
,”
J. Sound Vib.
,
240
(
4
), pp.
613
622
.10.1006/jsvi.2000.3252
30.
Yaman
,
M.
, and
Sen
,
S.
,
2007
, “
Vibration Control of a Cantilever Beam of Varying Orientation
,”
Int. J. Solids Struct.
,
44
(
3
), pp.
1210
1220
.10.1016/j.ijsolstr.2006.06.015
31.
Lotfazar
,
A.
,
Eghtesad
,
M.
, and
Najafi
,
A.
,
2008
, “
Vibration Control and Trajectory Tracking for General in-Plane Motion of an Euler–Bernoulli Beam Via Two-Time Scale and Boundary Control Methods
,”
ASME J. Vib. Acoust.
,
130
(
5
), p. 051009.10.1115/1.2948406
32.
Smyshlyaev
,
A.
,
Guo
,
B. Z.
, and
Krstic
,
M.
,
2009
, “
Arbitrary Decay Rate for Euler–Bernoulli Beam by Backstepping Boundary Feedback
,”
IEEE Trans. Autom. Control
,
54
(
5
), pp.
1134
1140
.10.1109/TAC.2009.2013038
33.
Dadfarnia
,
M.
,
Jalili
,
N.
,
Xian
,
B.
, and
Dawson
,
D. M.
,
2004
, “
Lyapunov-Based Vibration Control of Translational Euler–Bernoulli Beams Using the Stabilizing Effect of Beam Damping Mechanisms
,”
J. Vib. Control
,
10
(
7
), pp.
933
961
.10.1177/1077546304042070
34.
Reddy
,
J. N.
,
1986
,
Applied Functional Analysis and Variational Methods in Engineering
,
McGraw-Hill
,
New York
.
35.
Krstic
,
M.
, and
Smyshlyaev
,
A.
,
2008
,
Boundary Control of PDEs: A Course on Backstepping Designs
,
Society for Industrial Mathematics
,
Philadelphia, PA
.
36.
Huebner
,
K. H.
,
Dewhirst
,
D. L.
,
Smith
,
D. E.
, and
Byrom
,
T. G.
,
2008
,
The Finite Element Method for Engineers
,
Wiley
,
New York
.
37.
Pons
,
J.
,
2005
,
Emerging Actuator Technologies: A Micromechatronic Approach
,
Wiley
,
Chichester, UK
.
38.
Beaulieu
,
L. Y.
,
Godin
,
M.
,
Laroche
,
O.
,
Tabard-Cossa
,
V.
, and
Grütter
,
P.
,
2007
, “
A Complete Analysis of the Laser Beam Deflection Systems Used in Cantilever-Based Systems
,”
Ultramicroscopy
,
107
(
4
), pp.
422
430
.10.1016/j.ultramic.2006.11.001
You do not currently have access to this content.