A simple approach is proposed that can be used to analyze the free and forced responses of a combined system, consisting of an arbitrarily supported continuous structure carrying any number of lumped attachments. The assumed modes method is utilized to formulate the equations of motion, which conveniently leads to a form that allows one to exploit the Sherman-Morrison or the Sherman-Morrison-Woodbury formulas to compute the natural frequencies and frequency response of the combined system. Rather than solving a generalized eigenvalue problem to obtain the natural frequencies of the system, a frequency equation is formulated whose solution can be easily solved either numerically or graphically. In order to determine the response of the structure to a harmonic input, a method is formulated that leads to a reduced matrix whose inverse yields the same result as the traditional method, which requires the inversion of a larger matrix. The proposed scheme is easy to code, computationally efficient, and can be easily modified to accommodate arbitrarily supported continuous linear structures that carry any number of miscellaneous lumped attachments.

1.
Özgüven
,
H. N.
, and
Çandir
,
B.
, 1986, “
Suppressing the First and Second Resonances of Beams by Dynamic Vibration Absorbers
,”
J. Sound Vib.
0022-460X,
111
, pp.
377
390
.
2.
Cha
,
P. D.
, and
Wong
,
W. C.
, 1999, “
A Novel Approach to Determine the Frequency Equations of Combined Dynamical Systems
,”
J. Sound Vib.
0022-460X,
219
, pp.
689
706
.
3.
Gürgöze
,
M.
, 1996, “
On the Eigenfrequencies of a Cantilever Beam with Attached Tip Mass and a Spring-Mass System
,”
J. Sound Vib.
0022-460X,
190
, pp.
149
162
.
4.
Posiadała
,
B.
, 1997, “
Free Vibrations of Uniform Timoshenko Beams With Attachments
,”
J. Sound Vib.
0022-460X,
204
, pp.
359
369
.
5.
Lueschen
,
G. G. G.
,
Bergman
,
L. A.
, and
McFarland
,
D. M.
, 1996, “
Green’s Functions for Uniform Timoshenko Beams
,”
J. Sound Vib.
0022-460X,
194
, pp.
93
102
.
6.
Kukla
,
S.
, 1997, “
Application of Green Functions in Frequency Analysis of Timoshenko Beams with Oscillators
,”
J. Sound Vib.
0022-460X,
205
, pp.
355
363
.
7.
Chang
,
T. P.
,
Chang
,
F. I.
, and
Liu
,
M. F.
, 2001, “
On the Eigenvalues of a Viscously Damped Simple Beam Carrying Point Masses and Springs
,”
J. Sound Vib.
0022-460X,
240
, pp.
769
778
.
8.
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.
0022-460X,
136
, pp.
201
213
.
9.
Wu
,
J. S.
, and
Chou
,
H. M.
, 1999, “
A new Approach for Determining the Natural Frequencies and Mode Shapes of a Uniform Beam Carrying Any Number of Sprung Masses
,”
J. Sound Vib.
0022-460X,
220
, pp.
451
468
.
10.
Henderson
,
H. V.
, and
Searle
,
S. R.
, 1981, “
On Deriving the Inverse of a Sum of Matrices
,”
SIAM Rev.
0036-1445,
23
, pp.
53
60
.
11.
Hager
,
W. W.
, 1989, “
Updating the Inverse of a Matrix
,”
SIAM Rev.
0036-1445,
31
, pp.
221
239
.
12.
Ozer
,
M. B.
, and
Royston
,
T. J.
, 2005, “
Application of Sherman-Morrison Matrix Inversion Formula to Damped Vibration Absorbers Attached to Multi-Degree of Freedom Systems
,”
J. Sound Vib.
0022-460X,
283
, pp.
1235
1249
.
13.
Meirovitch
,
L.
, 2001,
Fundamentals of Vibrations
,
McGraw-Hill
,
New York
.
14.
Sherman
,
J.
, and
Morrison
,
W. J.
, 1949, “
Adjustment of an Inverse Matrix Corresponding to Changes in the Elements of a Given Column Or a Given Row of the Original Matrix
,”
Ann. Math. Stat.
0003-4851,
20
, p.
621
.
15.
Golub
,
G. H.
, and
van Loan
,
C. F.
, 1996,
Matrix Computations
,
Johns Hopkins University Press
,
Baltimore
.
16.
Gürgöze
,
M.
, 1998, “
On the Sensitivities of the Eigenvalues of a Viscously Damped Cantilever Carrying a Tip Mass
,”
J. Sound Vib.
0022-460X,
216
, pp.
215
225
.
17.
Wang
,
B. P.
,
Kitis
,
L.
,
Pilkey
,
W. D.
, and
Palazzolo
,
A.
, 1982, “
Structural Modifications to Achieve Antiresonance in Helicopters
,”
J. Aircr.
0021-8669,
19
, pp.
499
504
.
18.
Wang
,
B. P.
,
Kitis
,
L.
,
Pilkey
,
W. D.
, and
Palazzolo
,
A.
, 1985, “
Synthesis of Dynamic Vibration Absorbers
,”
ASME J. Vib., Acoust., Stress, Reliab. Des.
0739-3717,
107
, pp.
161
166
.
19.
Cha
,
P. D.
, 2004, “
Imposing Nodes at Arbitrary Locations for General Elastic Structures During Harmonic Excitations
,”
J. Sound Vib.
0022-460X,
272
, pp.
853
868
.
20.
Wang
,
B. P.
, 1993, “
Eigenvalue Sensitivity With Respect to Location of Internal Stiffness and Mass Attachments
,”
AIAA J.
0001-1452,
31
, pp.
791
794
.
21.
Cha
,
P. D.
, and
Zhou
,
X.
, 2006, “
Imposing Points of Zero Displacements and Zero Slopes Along Any Linear Structure During Harmonic Excitations
,”
J. Sound Vib.
0022-460X,
297
, pp.
55
71
.
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