Issue Section:
Research Papers
Abstract
This study revisits the free vibration of a cantilever beam with a mass-spring system attached at the tip of the beam. We recover the natural frequencies that were reported in some special case, and establish that there could be an additional, “lost” frequency that was somehow not uncovered in previous investigations. This frequency is found hereinafter.
Issue Section:
Research Papers
References
1.
Gershgorin
,
S.
, 1933
, “
Vibrations of Plates Loaded by Concentrated Masses
,” Prikladnaya Matematika i Mekhanika
,
1
, pp. 25
–37
.(in Russian).2.
Ormondroyd
,
J.
, and
Den Hartog
,
J. P.
, 1928
, “
The Theory of the Dynamic Vibration Absorber
,” Trans. Am. Soc. Mech. Eng.
,
50
, pp. 9
–22
.10.1115/1.40585533.
Young
,
D.
, 1952
, “
Theory of Dynamic Vibration Absorbers for Beams
,” Proceedings of the First U.S. National Congress of Applied Mechanics
,
ASME Press
,
New York
, pp. 91
–96
.4.
Snowdon
,
J. C.
, 1966
, “
Vibration of Cantilever Beams to Which Dynamic Absorbers Are Attached
,” J. Acoust. Soc. Am.
,
39
(5A
), pp. 878
–886
.10.1121/1.19099665.
Jacquot
,
R. G.
, and
Soedel
,
W.
, 1970
, “
Vibrations of Elastic Surface Systems Carrying Dynamic Elements
,” J. Acoust. Soc. Am.
,
47
(5B
), pp. 1354
–1358
.10.1121/1.19120426.
Jacquot
,
R. G.
, 1978
, “
Optimal Dynamic Vibration Absorbers for General Beam Systems
,” J. Sound Vib.
,
60
(4
), pp. 535
–542
.10.1016/S0022-460X(78)80090-X7.
Lee
,
T. W.
, 1973
, “
Vibration Frequency for a Uniform Beam With One End Spring Hinged and Carrying a Mass at the Other Free End
,” ASME J. Appl. Mech.
,
40
(3
), pp. 813
–815
.10.1115/1.34230978.
Parnell
,
L. A.
, and
Cobble
,
M. H.
, 1976
, “
Lateral Displacements of a Vibrating Cantilever Beam With a Concentrated Mass
,” J. Sound Vib.
,
44
(4
), pp. 499
–511
.10.1016/0022-460X(76)90092-49.
Bhat
,
R. B.
, and
Wagner
,
H.
, 1976
, “
Natural Frequencies of a Uniform Cantilever With a Tip Mass Slender in the Axial Direction
,” J. Sound Vib.
,
45
(2
), pp. 304
–307
.10.1016/0022-460X(76)90606-410.
Laura
,
P. A. A.
,
Pombo
,
J. L.
, and
Susemihl
,
E. A.
, 1974
, “
A Note on the Vibrations of a Clamped-Free Beam With a Mass at the Free End
,” J. Sound Vib.
,
37
(2
), pp. 161
–168
.10.1016/S0022-460X(74)80325-111.
Laura
,
P. A. A.
,
Susemihl
,
E. A.
,
Pombo
,
J. L.
,
Luisoni
,
L. E.
, and
Gelos
,
R.
, 1977
, “
On the Dynamic Behaviour of Structural Elements Carrying Elastically Mounted, Concentrated Masses
,” Appl. Acoust.
,
10
(2
), pp. 121
–145
.10.1016/0003-682X(77)90021-412.
Satter
,
M. A.
, 1979
, “
Vibration of Beams Carrying Discrete Dampers and Masses
,” ASME J. Mech. Des.
,
101
(2
), pp. 317
–321
.10.1115/1.345405613.
Takahashi
,
K.
, 1980
, “
Eigenvalue Problem of a Beam With a Mass and Spring at the End Subjected to an Axial Force
,” J. Sound Vib.
,
71
(3
), pp. 453
–457
.10.1016/0022-460X(80)90427-714.
Stephen
,
N. G.
, 1980
, “
Vibration of a Cantilevered Beam Carrying a Tip Heavy Body by Dunkerley's Method
,” J. Sound Vib.
,
70
(3
), pp. 463
–465
.10.1016/0022-460X(80)90315-615.
Grossi
,
R. O.
, and
Laura
,
P. A. A.
, 1982
, “
Further Results on a Vibrating Beam With a Mass and Spring at the End Subjected to an Axial Force
,” J. Sound Vib.
,
84
(4
), pp. 593
–594
.10.1016/S0022-460X(82)80039-416.
Lau
,
J. H.
, 1984
, “
Vibration Frequencies and Mode Shapes for a Constrained Cantilever
,” ASME J. Appl. Mech.
,
51
(1
), pp. 182
–187
.10.1115/1.316756517.
Gürgöze
,
M.
, 1984
, “
A Note on the Vibrations of Restrained Beams and Rods With Point Masses
,” J. Sound Vib.
,
96
(4
), pp. 461
–468
.10.1016/0022-460X(84)90633-318.
Gürgöze
,
M.
, 1985
, “
On the Vibrations of Restrained Beams and Rods With Heavy Masses
,” J. Sound Vib.
,
100
(4
), pp. 588
–589
.10.1016/S0022-460X(85)80009-219.
Laura
,
P. A. A.
,
Filipich
,
C. P.
, and
Cortinez
,
V. H.
, 1987
, “
Vibrations of Beams and Plates Carrying Concentrated Masses
,” J. Sound Vib.
,
117
(3
), pp. 459
–465
.10.1016/S0022-460X(87)80065-220.
Ercoli
,
L.
, and
Laura
,
P. A. A.
, 1987
, “
Analytical and Experimental Investigation on Continuous Beams Carrying Elastically Mounted Masses
,” J. Sound Vib.
,
114
(3
), pp. 519
–533
.10.1016/S0022-460X(87)80021-421.
Liu
,
W. H.
, and
Huang
,
C. C.
, 1988
, “
Vibrations of a Constrained Beam Carrying a Heavy Tip Body
,” J. Sound Vib.
,
123
(1
), pp. 15
–29
.10.1016/S0022-460X(88)80074-922.
Wu
,
J. S.
, and
Lin
,
T. L.
, 1990
, “
Free Vibration Analysis of a Uniform Cantilever Beam With Point Masses by an Analytical-and-Numerical-Combined Method
,” J. Sound Vib.
,
136
(2
), pp. 201
–213
.10.1016/0022-460X(90)90851-P23.
Hamdan
,
M. N.
, and
Jubran
,
B. A.
, 1991
, “
Free and Forced Vibrations of a Restrained Uniform Beam Carrying an Intermediate Lumped Mass and a Rotary Inertia
,” J. Sound Vib.
,
150
(2
), pp. 203
–216
.10.1016/0022-460X(91)90616-R24.
Larrondo
,
H.
,
Avalos
,
D. L.
, and
Laura
,
P. A. A.
, 1992
, “
Natural Frequencies of a Bernoulli Beam Carrying an Elastically Mounted Concentrated Mass
,” Ocean Eng.
,
19
(5
), pp. 461
–468
.10.1016/0029-8018(92)90004-N25.
Aida
,
T.
,
Toda
,
S.
,
Ogawa
,
N.
, and
Imada
,
Y.
, 1992
, “
Vibration Control of Beams by Beam-Type Dynamic Vibration Absorbers
,” J. Eng. Mech.
,
118
(2
), pp. 248
–258
.10.1061/(ASCE)0733-9399(1992)118:2(248)26.
Jen
,
M. U.
, and
Magrab
,
E. B.
, 1993
, “
Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System
,” ASME J. Vib. Acoust.
,
115
(2
), pp. 202
–209
.10.1115/1.293033127.
Rossi
,
R. E.
,
Laura
,
P. A. A.
,
Avalos
,
D. R.
, and
Larrondo
,
H. A.
, 1993
, “
Free Vibrations of Timoshenko Beams Carrying Elastically Mounted, Concentrated Masses
,” J. Sound Vib.
,
165
(2
), pp. 209
–223
.10.1006/jsvi.1993.125428.
Pesterev
,
A. V.
, and
Tavrizov
,
G. A.
, 1994
, “
On Vibrations of Beams With Oscillators, I: Structural Analysis Method for Solving the Spectral Problem
,” J. Sound Vib.
,
170
(4
), pp. 521
–536
.10.1006/jsvi.1994.108129.
Pesterev
,
A. V.
, and
Bergman
,
L. A.
, 1995
, “
On Vibration of a System With an Eigenfrequency Identical to That of One of Its Subsystems
,” ASME J. Vib. Acoust.
,
117
(4
), pp. 482
–487
.10.1115/1.287448730.
Gürgöze
,
M.
, 1996
, “
On the Eigenfrequencies of a Cantilever Beam With Attached Tip Mass and a Spring-Mass System
,” J. Sound Vib.
,
190
(2
), pp. 149
–162
.10.1006/jsvi.1996.005331.
Andreev
,
L. V.
,
Dyshko
,
A. L.
, and
Pavlenko
,
I. D.
, 2000
, Dynamics of Plates and Shells With Concentrated Masses
, “
Mashinostroenie” Publishing House
,
Moscow
(in Russian).32.
Rossit
,
C. A.
, and
Laura
,
P. A. A.
, 2001
, “
Free Vibrations of a Cantilever Beam With a Spring–Mass System Attached to the Free End
,” Ocean Eng.
,
28
(7
), pp. 933
–939
.10.1016/S0029-8018(00)00055-X33.
Bambill
,
D. V.
, and
Rossit
,
C. A.
, 2002
, “
Forced Vibrations of a Beam Elastically Restrained Against Rotation and Carrying a Spring–Mass System
,” Ocean Eng.
,
29
(6
), pp. 605
–626
.10.1016/S0029-8018(01)00042-734.
Cha
,
P. D.
, 2005
, “
A General Approach to Formulating the Frequency Equation for a Beam Carrying Miscellaneous Attachments
,” J. Sound Vib.
,
286
(4–5
), pp. 921
–939
.10.1016/j.jsv.2004.10.01235.
Lin
,
H.-Y.
, and
Tsai
,
Y.-C.
, 2007
, “
Free Vibration Analysis of a Uniform Multi-Span Beam Carrying Multiple Spring-Mass Systems
,” J. Sound Vib.
,
302
(3
), pp. 442
–456
.10.1016/j.jsv.2006.06.08036.
Banerjee
,
J. R.
, 2012
, “
Free Vibration of Beams Carrying Spring-Mass Systems-A Dynamic Stiffness Approach
,” Comput. Struct.
,
104–105
, pp. 21
–26
.10.1016/j.compstruc.2012.02.02037.
Wang
,
Z.
,
Hong
,
M.
,
Xu
,
J.-C.
, and
Cui
,
H.-G.
, 2014
, “
Analytical and Experimental Study of Free Vibration of Beams Carrying Multiple Masses and Springs
,” J. Mar. Sci. Appl.
,
13
(1
), pp. 32
–40
.10.1007/s11804-014-1231-438.
Kumar
,
S. V.
,
Sarkar
,
A.
, and
Padmanabhan
,
C.
, 2019
, “
Free Vibration of Spring Monted-Beams, IOP Conference Series
,” J. Phys.: Conf. Ser.
,
1264
, p. 012026
.10.1088/1742-6596/1264/1/012026Copyright © 2024 by ASME
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