The problem considered is that of solving for the control input that generates partly specified motion of a deformable structure with distributed piezoelectric actuation. The motion constraint, called the program constraint, is specified as a desired relation on the motion of selected material points of the structure. The solution is based on a projection method applicable to a class of finite-dimensional dynamical systems which includes many common vibration models. For a nonlinear model with a nonlinear program constraint, the procedure in general results in a set of differential algebraic equations. It is shown that for linear models with linear periodic program constraints, the system is reduced to a set of algebraic equations. Application examples are presented for a Euler-Bernoulli beam to demonstrate the usefulness of the procedure.

References

References
1.
Blajer
,
W.
, and
Kolodziejczyk
,
K.
, 2004, “
A Geometric Approach to Solving Problems of Control Constraints: Theory and DAE Framework
,”
Multibody Syst. Dyn.
,
11
,
343
364
.
2.
Chen
,
Y -H.
, 2004, “
Mechanical Systems Under Servo Constraints: The Lagrange’s Approach
,”
Mechatronics
,
15
,
317
337
.
3.
Parczewski
,
J.
, and
Blajer
,
W.
, 1989, “
On Realization of Program Constraints: Part I—Theory, Part II—Practical Implications
,”
J. Appl. Mech.
,
56
,
676
684
.
4.
Blajer
,
W.
, 1992, “
A Projection Method Approach to Constrained Dynamic Analysis
,”
Appl. Mech.
,
59
,
643
649
.
5.
Blajer
,
W.
, 1993, “
Structure of Differential-Algebraic Equations for Control Problems in Mechanics
,”
Int. J. Syst. Sci.
,
24
,
2367
2377
.
6.
Blajer
,
W.
, 1997, “
Dynamics and Control of Mechanical Systems in Partly Specified Motion
,”
J. Franklin Inst.
,
334B
,
407
426
.
7.
Rosen
,
A.
, 1999, “
Applying the Lagrange Method to Solve Problems of Control Constraints
,”
J. Appl. Mech.
,
66
,
1013
1015
.
8.
Chen
,
Y -H.
, 2007, “
Equations of Motion of Mechanical Systems Under Servo Constraints: The Maggi Approach
,”
Mechatronics
,
18
,
208
217
.
9.
Kolodziejczyk
,
K.
, and
Blajer
,
W.
, 2007, “
Control of Underactuated Mechanical Systems With Servo-Constraints
,”
Nonlinear Dyn.
,
50
,
781
791
.
10.
Dziewiecki
,
W.
,
Kolodziejczyk
,
K.
,
Blajer
,
W.
, and
Mazur
,
Z.
, 2010, “
Inverse Dynamics of Underactuated Mechanical Systems: A Simple Case Study and Experimental Verification
,”
J. Commun. Nonlin. Sci. Numer. Sim.
,
16
,
2265
2272
.
11.
Kane
,
T. R.
:
Levinson
,
D. A.
, 1985,
Dynamics: Theory and Applications
,
New York
,
McGraw-Hill
.
12.
Huston
,
R. L.
, 1990,
Multibody Dynamics
,
London
,
Butterworth-Heinemann
.
13.
Sanders
,
B.
,
Eastep
,
F. E.
, and
Foster
,
E.
, 2007, “
Aerodynamic and Aeroelastic Considerations of Wings With Conformal Control Surfaces for Morphing Aircraft
,”
J. Aircraft
,
40
,
94
99
.
14.
Sashida
,
T.
, and
Kenjo
,
T.
, 1993,
An Introduction to Ultrasonic Motors
,
Oxford University Press
,
Oxford, UK
.
15.
Zhao
,
H.
,
Wu
,
J.
, and
Luo
,
J. S.
, 2004, “
Turbulent Drag Reduction by Traveling Wave of Flexible Wall
,”
Fluid Dyn. Res.
,
34
,
175
198
.
16.
Shabana
,
A. A.
, 1997, “
Flexible Multibody Dynamics: Review of Past and Recent Developments
,”
Multibody Syst. Dyn.
,
1
,
189
222
.
17.
Crawley
,
E. F.
, 1994, “
Intelligent Structures for Aerospace: A Technology Overview and Assessment
,”
AIAA J.
,
32
(
8
),
1689
1699
.
18.
Crawley
,
E. F.
, and
de Luis
,
J.
, 1987, “
Use of Piezoelectric Actuators as Elements of Intelligent Structures
,”
AIAA J.
,
25
(
10
),
1373
1385
.
19.
Irschik
,
H.
, 2002, “
A Review on Static and Dynamic Shape Control of Structures by Piezoelectric Actuation
,”
Eng. Struct.
,
24
(
1
),
5
11
.
20.
Lam
,
S. H.
, 1998, “
On Lagrangian Dynamics and Its Control Formulation
,”
Appl. Math. Comput.
,
91
,
259
284
.
21.
Tzou
,
H. S.
, 1993,
Piezoelectric Shells: Distributed Sensing and Control of Continua
,
Berlin
,
Springer
.
22.
Meirovitch
,
L.
, 1996,
Principles and Techniques of Vibrations
,
Englewood Cliffs
,
NJ, Prentice Hall
.
23.
Tzou
,
H. S.
, 1991, “
Distributed Modal Identification and Vibration Control of Continua: Theory and Applications
,”
J. Dyn. Syst., Meas., Control
,
113
,
494
499
.
You do not currently have access to this content.