This paper discusses the control of coupled bending and axial vibrations in L-shaped and portal planar frame structures. The controller is designed based on a wave standpoint, in which vibrations are described as waves traveling along uniform structural waveguides, and being reflected and transmitted at structural discontinuities. Active discontinuities are created using active control forces/moments both along structural elements and at structural joints to control vibration waves. The classical Euler–Bernoulli as well as the advanced Timoshenko bending theories are applied in modeling and controlling the flexural vibrations in planar frames. The axial vibrations are modeled using the elementary theory as it is typically valid for frequencies up to twice the cutoff frequency of Timoshenko bending waves. Results are compared between the two bending vibration theories. It is concluded that for relatively higher frequencies, typically when the transverse dimensions are not negligible with respect to the wavelength, the effects of rotary inertia and shear distortion must be taken into account for both vibration analysis and control design.

References

References
1.
Chang
,
C. H.
,
1978
, “
Vibrations of Frames With Inclined Members
,”
J. Sound Vib.
,
56
(
2
), pp.
201
214
.10.1016/S0022-460X(78)80015-7
2.
Lin
,
H. P.
, and
Ro
,
J.
,
2003
, “
Vibration Analysis of Planar Serial-Frame Structures
,”
J. Sound Vib.
,
262
(
2
), pp.
1113
1131
.10.1016/S0022-460X(02)01089-1
3.
Mei
,
C.
,
2012
, “
Wave Analysis of In-Plane Vibrations of L-Shaped and Portal Planar Frame Structures
,”
ASME J. Vibr. Acoust.
,
134
, p.
021011
.10.1115/1.4005014
4.
Graff
,
K. F.
,
1975
,
Wave Motion in Elastic Solids
,
Ohio State University Press
, Columbus, OH.
5.
Cremer
,
L.
,
Heckl
,
M.
, and
Ungar
,
E. E.
,
1987
,
Structure-Borne Sound
,
Springer-Verlag
,
Berlin
.
6.
Doyle
,
J. F.
,
1989
,
Wave Propagation in Structures
,
Springer-Verlag
,
New York
.
7.
Rayleigh
,
L.
,
1926
,
Theory of Sound
,
The Macmillan Company
,
New York
.
8.
Timoshenko
,
S. P.
,
1921
, “
On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars
,”
Philos. Mag.
,
41
(
245
), pp.
744
746
.10.1080/14786442108636264
9.
Timoshenko
,
S. P.
,
1922
, “
Transverse Vibrations of Bars of Uniform Cross Section
,”
Philos. Mag.
,
43
(
253
), pp.
125
131
.10.1080/14786442208633855
10.
Wang
,
C. H.
, and
Rose
,
L. R. F.
,
2003
, “
Wave Reflection and Transmission in Beams Containing Delamination and Inhomogeneity
,”
J. Sound Vib.
,
264
, pp.
851
872
.10.1016/S0022-460X(02)01193-8
11.
Mace
,
B. R.
,
1987
, “
Active Control of Flexural Vibrations
,”
J. Sound Vib.
,
114
(
2
), pp.
253
270
.10.1016/S0022-460X(87)80152-9
12.
Lu
,
J.
,
Crocker
,
M. J.
, and
Raju
,
P. K.
,
1989
, “
Active Vibration Control Using Wave Control Concepts
,”
J. Sound Vib.
,
134
(
2
), pp.
364
368
.10.1016/0022-460X(89)90659-7
13.
Elliott
,
S. J.
, and
Billet
,
L.
,
1993
, “
Adaptive Control of Flexural Waves Propagating in a Beam
,”
J. Sound Vib.
,
163
(
2
), pp.
295
310
.10.1006/jsvi.1993.1166
14.
Vipperman
,
J. S.
,
Burdisso
,
R. A.
, and
Fuller
,
C. R.
,
1993
, “
Active Control of Broadband Structural Vibration Using the LMS Adaptive Algorithm
,”
J. Sound Vib.
,
166
(
2
), pp.
283
299
.10.1006/jsvi.1993.1297
15.
Fuller
,
C. R.
,
Gibbs
,
G. P.
, and
Silcox
,
R. J.
,
1990
, “
Simultaneous Active Control of Flexural and Extensional Waves in Beams
,”
J. Intell. Mater. Syst. Struct.
,
1
, pp.
235
247
.10.1177/1045389X9000100206
16.
Clark
,
R. L.
,
Pan
,
J.
, and
Hansen
,
C. H.
,
1992
, “
An Experimental Study of the Active Control of Multiple-Wave Types in an Elastic Beam
,”
J. Acoust. Soc. Am.
,
92
(
2
), pp.
871
876
.10.1121/1.403957
17.
Von Flotow
,
A. H.
,
1985
, “
Traveling Wave Control for Large Spacecraft Structures
,”
J. Guid. Control
,
9
(
4
), pp.
462
468
.10.2514/3.20133
18.
Gardonio
,
P.
, and
Elliott
,
S. J.
,
1995
, “
Active Control of Multiple Waves Propagating on a One-Dimensional System With a Scattering Termination
,” Active 95: International Symposium on Active Control of Sound and Vibration, Newport Beach, CA, July 6–8, pp. 115–126.
19.
Mace
,
B. R.
, and
Jones
,
R. W.
,
1996
, “
Feedback Control of Flexural Waves in Beams
,”
J. Struct. Control
,
3
(
1–2
), pp.
89
98
.10.1002/stc.4300030108
20.
Elliott
,
S. J.
,
Brennan
,
M. J.
, and
Pinnington
,
R. J.
,
1993
, “
Feedback Control of Flexural Waves on a Beam
,”
Proceedings of Inter-Noise 93
,
Leuven, Belgium
, August 24–26, pp.
843
846
.
21.
Brennan
,
M. J.
,
1994
, “
Active Control of Waves on One-Dimensional Structure
,”
Ph.D. thesis
,
University of Southampton
, Southampton, UK.
22.
Mei
,
C.
,
Mace
,
B. R.
, and
Jones
,
R. W.
,
2001
, “
Hybrid Wave/Mode Active Vibration Control
,”
J. Sound Vib.
,
247
(
5
), pp.
765
784
.10.1006/jsvi.2001.3795
23.
Mei
,
C.
,
2009
, “
Hybrid Wave/Mode Active Vibration Control of Bending Vibrations in Beams Based on Advanced Timoshenko Theory
,”
J. Sound Vib.
,
322
(
1–2
), pp.
29
38
.10.1016/j.jsv.2008.11.003
24.
Mei
,
C.
,
2011
, “
Wave Control of Vibrations in Multi-Story Planar Frame Structures Based on Classical Vibration Theories
,”
J. Sound Vib.
,
330
, pp.
5530
5544
.10.1016/j.jsv.2011.06.022
25.
Ginsberg
,
J. H.
,
2001
,
Mechanical and Structural Vibrations
,
Wiley
,
New York
.
26.
Mei
,
C.
,
2008
, “
Wave Analysis of In-Plane Vibrations of H- and T-Shaped Planar Frame Structures
,”
ASME J. Vibr. Acoust.
,
130
, p.
061004
.10.1115/1.2980373
27.
Cowper
,
G. R.
,
1966
, “
The Shear Coefficient in Timoshenko's Beam Theory
,”
ASME J. Appl. Mech.
,
33
, pp.
335
340
.10.1115/1.3625046
28.
Mei
,
C.
,
2010
, “
In-Plane Vibrations of Classical Planar Frame Structures—An Exact Wave-Based Analytical Solution
,”
J. Vib. Control
,
16
(
9
), pp.
1265
1285
.10.1177/1077546309339422
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