In this work we consider segmented Euler-Bernoulli beams that can have an internal damping of the type Kelvin-Voight and external viscous damping at the discontinuities of the sections. In the literature, the study of this kind of beams has been sufficiently studied with proportional damping only, however the effects of non-proportional damping has been little studied in terms of modal analysis. The obtaining of the modes of segmented beams can be accomplished with a the state space methodology or with the classical Euler construction of responses. Here, we follow a newtonian approach with the use of the impulse response of beams subject both types of damping. The use of the dynamical basis, generated by the fundamental solution of a differential equation of fourth order, allows to formulate the eigenvalue problem and the shapes of the modes in a compact manner. For this, we formulate in a block manner the boundary conditions and intermediate conditions at the beam and values of the fundamental matrix at the ends of the beam and in the points intermediate. We have chosen a basis generated by a fundamental response and it derivatives. The elements of this basis has the same shape with a convenient translation for each segment. This choice reduce computations with the number of constants to be determined to find only the ones that correspond to the first segment. The eigenanalysis will allow to study forced responses of multi-span Euler-Bernoulli beams under classical and non-classical boundary conditions as well as multi-walled carbon nanotubes (MWNT) that are modelled as an assemblage of Euler-Bernoulli beams connected throughout their length by springs subject to van der Waals interaction between any two adjacent nanotubes.

1.
Gorman D., 1975. Free Vibration Analysis of Beams and Shafts, John Wiley.
2.
Jang
S. K.
,
Bert
C. W.
.
Free Vibrations of Stepped Beams: Exact and Numerical Solutions
,
1989
.
Journal of Sound and Vibration
, vol.
130
(
2)
,
342
346
.
3.
Naguleswara
S.
.
Lateral Vibration of a Uniform Euler-Bernoulli Beam Carrying a Particle at an Intermediate Point
1999
.
Journal of Sound and Vibration
, vol.
43
,
2737
2752
.
4.
De Rosa
M. A.
,
Belles
P. M.
,
Maurizi
M. J.
.
Free Vibrations of Stepped Beams with Intermediate Elastic Supports
1995
.
Journal of Sound and Vibration
, vol.
181
(
3)
,
905
910
.
5.
Vu
H. V.
,
Ordon˜ez
A. M.
,
Karnopp
B. H.
.
Vibration of a Double Beam System
2000
.
Journal of Sound and Vibration
. vol.
229
(
4)
,
807
822
.
6.
Turhan
O¨.
.
On the Eigencharacteristics of Longitudinally Vibrating Rods with a Cross-Section Discontinuity
2001
.
Journal of Sound and Vibration
, vol.
248
(
1)
,
167
177
.
7.
Korenev, B.G., Reznikov, L.M. 1993. Dynamic Vibration Absorbers, John Wiley & Sons.
8.
Krylov, A.N. 1948. Collected works, Vol.10, Ship Vibration, Moscow, Lenningrad, ANSSSR.
9.
De Rosa
M. A.
.
Free Vibrations of Stepped Beams with Elastic Ends
1994
.
Journal of Sound and Vibration
, vol.
173
(
4)
,
563
567
.
10.
Us`cilowska
A.
,
Kolodziej
J. A.
.
Free Vibrations of Immersed Column Carrying a Tip Mass
1998
.
Journal of Sound and Vibration
, vol.
216
(
1)
,
147
157
.
11.
Lee
S. Y
,
Ke
H. Y.
.
Free Vibrations of Non-Uniform Beams Resting on Non-Uniform Elastic Foundation with General Elastic End Restraints
1990
.
Computers and Structures
, vol.
14
(
3)
,
421
429
.
12.
Low
K. H.
.
On the Method to Derive Frequency Equations of Beams Carrying Multiple Masses
2001
.
Int. Journal of Mechanical Sciences
, vol.
43
,
871
881
.
13.
Tsukazan
T.
,
The Use of a Dynamical Basis for Computing the Modes of a Beam System with a Discontinuous Cross-Section
2005
.
Journal of Sound and Vibration
, vol.
281
, pp.
1175
1185
.
14.
Claeyssen
J. R.
,
Suazo
G. C.
,
Jung
C.
.
A Direct Approach to Second-Order Matrix Non-Classical Vibrating Equations
1999
.
Applied Numerical Mathematics
, vol.
39
,
65
78
.
15.
Claeyssen
J. R.
,
Soder
R. A.
.
A Dynamical Basis for Computing Modes of Euler-Bernoulli and Timoshenko Beams
2003
.
Journal of Sound and Vibration
, vol.
259
(
4)
,
986
990
.
16.
Claeyssen, J.R., Tsukazan, T. 1990. Quart.App.Math. Dynamic Solutions of Linear Matrix Differential Equations, vol. XLVIII (1).
17.
Sorrentino
S.
,
Marchesiello
S.
,
Piombo
B. A. D.
.
A New Analytical Technique for Vibration Analysis of Non-Proportionally Damped Beams
2003
.
Journal of Sound and Vibration
, vol.
265
(
4)
,
765
782
.
18.
Ginsberg, J. 2002. Mechanical and Structural Vibrations. John Wiley & Sons.
19.
Clark, S.K. 1972. Dynamics of Continuous Elements. Prentice-Hall.
20.
Miller, K.S. 1963. Linear Differential Equations in the Real Domain. New York, W.W.Norton Co.
21.
Chakraboty, A., Sivakumur, M.S., Gopalakrishnan, S., Int. J. Solids and Structures, Spectral element based model for wave propagation analysis in multi-wall carbon nanotubes, (in press).
22.
Claeyssen
J. C.
,
Moraes
I. F.
,
Copetti
R. D.
.
Decomposition of Forced Responses in Vibrating Systems
2003
.
Applied Numerical Mathematics
, vol
47
,
391
405
.
23.
Leipholz, H., 1980. it Stability of Elastic Systems, Sitthoff & Noordoff, Netherlands.
24.
Han
S. M.
,
Benaroya
H.
,
Wei
T.
.
Dynamics of Transversally Vibrating Beams using Four Engineering Theories
Journal of Sound and Vibration
, vol.
225
(
5)
,
935
988
.
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