The numerical evaluation of high-order modes of a uniform Euler–Bernoulli beam has been studied by reformatting the classical expression of mode shapes. That method, however, is inapplicable to a stepped beam due to the nonuniform expressions of the mode shape for each beam segment. Given that concern, this study develops an alternative method for the numerical evaluation of high-order modes for stepped beams. This method effectively expands the space of high-order modal solutions by introducing local coordinate systems to replace the conventional global coordinate system. This set of local coordinate systems can significantly simplify the frequency determinant of vibration equations of a stepped beam, in turn, largely eliminating numerical round-off errors and conducive to high-order mode evaluation. The efficacy of the proposed scheme is validated using various models of Euler–Bernoulli stepped beams. The principle of the method has the potential for extension to other types of Euler–Bernoulli beams with discontinuities in material and geometry. (The Matlab code for the numerical evaluation of high-order modes for stepped beams can be provided by the corresponding author upon request.)

References

References
1.
Bai
,
R. B.
,
Cao
,
M. S.
,
Su
,
Z. Q.
,
Ostachowicz
,
W.
, and
Xu
,
H.
,
2012
, “
Fractal Dimension Analysis of Higher-Order Mode Shapes for Damage Identification of Beam Structures
,”
Math. Probl. Eng.
,
2012
, p.
454568
.10.1155/2012/454568
2.
Wei
,
G. W.
,
Zhao
,
Y. B.
, and
Xiang
Y.
,
2002
, “
A Novel Approach for the Analysis of High-Frequency Vibrations
,”
J. Sound Vib.
,
257
(
2
), pp.
207
246
.10.1006/jsvi.2002.5055
3.
Zhang
,
W. G.
,
Wang
,
A. M.
,
Vlahopoulos
,
N.
, and
Wu
,
K. C.
,
2003
, “
High-Frequency Vibration Analysis of Thin Elastic Plates Under Heavy Fluid Loading by an Energy Finite Element Formulation
,”
J. Sound Vib.
,
263
, pp.
21
46
.10.1016/S0022-460X(02)01096-9
4.
Lau.
E.
,
Al-Dujaili
,
S.
,
Guenther
,
A.
,
Liu
,
D.
,
Wang
,
L.
, and
You
,
L.
,
2010
, “
Effect of Low-Magnitude, High-Frequency Vibration on Osteocytes in the Regulation of Osteoclasts
,”
Bone
,
46
, pp.
1508
1515
.10.1016/j.bone.2010.02.031
5.
Tang
,
Y.
,
2003
, “
Numerical Evaluation of Uniform Beam Modes
,”
J. Eng. Mech.
,
12
, pp.
1475
1477
.10.1061/(ASCE)0733-9399(2003)129:12(1475)
6.
Goncalves
,
P. J. P.
,
Brennan
,
M. J.
, and
Elliott
,
S. J.
,
2007
, “
Numerical Evaluation of High-Order Modes of Vibration in Uniform Euler–Bernoulli Beams
,”
J. Sound Vib.
,
301
, pp.
1035
1039
.10.1016/j.jsv.2006.10.012
7.
Shankar
,
K.
, and
Keane
,
A. J.
,
1994
, “
Energy Flow Predictions in a Structure of Rigidly Joined Beams Using Receptance Theory
,”
J. Sound Vib.
,
185
, pp.
867
890
.10.1006/jsvi.1995.0422
8.
Low
,
K. H.
,
1993
, “
A Reliable Algorithm for Solving Frequency Equations Involving Transcendental Functions
,”
J. Sound Vib.
,
161
, pp.
369
377
.10.1006/jsvi.1993.1080
9.
Michael
,
A. K.
,
Abhijit
,
B.
, and
Brain
,
P. M.
,
2006
, “
Closed Form Solutions for the Dynamic Response of Euler–Bernoulli Beams With Step Changes in Cross Section
,”
J. Sound Vib.
,
295
, pp.
214
225
.10.1016/j.jsv.2006.01.008
10.
Naguleswaran
,
S.
,
2002
, “
Natural Frequencies Sensitivity and Mode Shape Details of an Euler–Bernoulli Beam With One-Step Change in Cross-Section and With Ends on Classical Supports
,”
J. Sound Vib.
,
254
(
4
), pp.
751
767
.10.1006/jsvi.2001.3743
11.
G.
Failla
,
2011
, “
Closed-Form Solutions for Euler–Bernoulli Arbitrary Discontinuous Beams
,”
Arch. Appl. Mech.
,
81
, pp.
605
608
.10.1007/s00419-010-0434-7
12.
Kukla
,
S.
, and
Zamojska
,
I.
,
2007
, “
Frequency Analysis of Axially Loaded Stepped Beams by Green's Function Method
,”
J. Sound Vib.
,
300
, pp.
1034
1041
.10.1016/j.jsv.2006.07.047
13.
Institute of Electrical and Electronics Engineers
,
2008
, “
754-2008-IEEE Standard for Floating-Point Arithmetic
,”
IEEE
,
NewYork
.
You do not currently have access to this content.