Abstract

Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

References

1.
Emam
,
S. A.
,
2009
, “
A Static and Dynamic Analysis of the Postbuckling of Geometrically Imperfect Composite Beams
,”
Compos. Struct.
,
90
(
2
), pp.
247
253
.10.1016/j.compstruct.2009.03.020
2.
Celebi
,
K.
,
Yarimpabuc
,
D.
, and
Tutuncu
,
N.
,
2018
, “
Free Vibration Analysis of Functionally Graded Beams Using Complementary Functions Method
,”
Arch. Appl. Mech.
,
88
(
5
), pp.
729
739
.10.1007/s00419-017-1338-6
3.
Marrec
,
L. L.
,
Lerbet
,
J.
, and
Rakotomanana
,
L. R.
,
2018
, “
Vibration of a Timoshenko Beam Supporting Arbitrary Large Pre-Deformation
,”
Acta Mech.
,
229
(
1
), pp.
109
132
.10.1007/s00707-017-1953-x
4.
Stoykov
,
S.
, and
Ribeiro
,
P.
,
2010
, “
Nonlinear Forced Vibrations and Static Deformations of 3D Beams With Rectangular Cross Section: The Influence of Warping, Shear Deformation and Longitudianl Displacements
,”
Int. J. Mech. Sci.
,
52
(
11
), pp.
1505
1521
.10.1016/j.ijmecsci.2010.06.011
5.
Lenci
,
S.
,
Clementi
,
F.
, and
Rega
,
G.
,
2016
, “
A Comprehensive Analysis of Hardening/Softening Behaviour of Shearable Planar Beams With Whatever Axial Boundary Constrain
,”
Meccanica
,
51
(
11
), pp.
2589
2606
.10.1007/s11012-016-0374-6
6.
Ding
,
H.
, and
Chen
,
L.-Q.
,
2019
, “
Nonlinear Vibration of a Slightly Curved Beam With Quasi-Zero-Stiffness Isolators
,”
Nonlinear Dyn.
,
95
(
3
), pp.
2367
2382
.10.1007/s11071-018-4697-9
7.
Ghayesh
,
M. H.
, and
Farokhi
,
H.
,
2018
, “
Bending and Vibration Analysis of Coupled Axially Functionally Graded Tappered Beams
,”
Nonlinear Dyn.
,
91
(
1
), pp.
17
28
.10.1007/s11071-017-3783-8
8.
Domagalski
,
L.
,
2018
, “
Free and Forced Large Amplitude Vibrations of Periodically Inhomogeneous Slender Beams
,”
Arch. Civ. Mech. Eng.
,
18
(
4
), pp.
1506
1519
.10.1016/j.acme.2018.06.005
9.
Sharma
,
P.
,
2018
, “
Efficacy of Harmonic Differential Quadrature Method to Vibration Analysis of FGPM Beam
,”
Compos. Struct.
,
189
, pp.
107
116
.10.1016/j.compstruct.2018.01.059
10.
Banerjee
,
J. R.
, and
Ananthapuvirajah
,
A.
,
2018
, “
Free Vibration of Functionally Graded Beams and Frameworks Using the Dynamic Stiffness Method
,”
J. Sound Vib.
,
422
, pp.
34
47
.10.1016/j.jsv.2018.02.010
11.
Ghayesh
,
M. H.
,
2012
, “
Nonlinear Dynamic Response of a Simply-Supported Kelvin-Voigt Viscoelastic Beam, Additionally Supported by a Nonlinear Spring
,”
Nonlinear Anal.: Real World Appl.
,
13
(
3
), pp.
1319
1333
.10.1016/j.nonrwa.2011.10.009
12.
Ghayesh
,
M. H.
,
2019
, “
Viscoelastic Dynamics of Axially FG Microbeams
,”
Int. J. Eng. Sci.
,
135
, pp.
75
85
.10.1016/j.ijengsci.2018.10.005
13.
Scurtu
,
P. R.
,
Clark
,
M.
, and
Zu
,
J. W.
,
2012
, “
Coupled Longitudinal and Transverse Vibration of Automotive Belts Under Longitudinal Excitations Using Analog Equation Method
,”
J. Vib. Control
,
18
(
9
), pp.
1336
1352
.10.1177/1077546311418866
14.
Wang
,
J.
,
Zhang
,
Z.
, and
Hua
,
H. X.
,
2016
, “
Coupled Flexural-Longitudianl Vibrations of Timenshenko Double-Beam Systems Introduced by Mass Eccentricities
,”
Int. J. Appl. Mech.
,
8
(
5
), p.
1650067
.10.1142/S1758825116500678
15.
Szekrenyes
,
A.
,
2014
, “
Coupled Flexural-Longitudianl Vibration of Delaminated Composite Beams With Local Stability Analysis
,”
J. Sound Vib.
,
333
(
20
), pp.
5141
5164
.10.1016/j.jsv.2014.05.021
16.
Chen
,
W. R.
, and
Chang
,
H.
,
2018
, “
Vibration Analysis of Functionally Graded Timenshenko Beams
,”
Int. J. Struct. Stability Dyn.
,
18
(
1
), p.
1850007
.10.1142/S0219455418500074
17.
Gebhardt
,
C. G.
,
Matusevich
,
A. E.
, and
Inaudi
,
J.
,
2018
, “
Coupled Transverse and Axial Vibrations Including Warping Effect in Asymmetric Short Beams
,”
J. Eng. Mech.
,
144
(
6
), p.
04018043
.10.1061/(ASCE)EM.1943-7889.0001471
18.
Zhu
,
K.
, and
Chung
,
J.
,
2019
, “
Vibration and Stability Analysis of a Simply-Supported Rayleigh Beam With Spinning and Axial Motions
,”
Appl. Math. Modell.
,
66
, pp.
362
382
.10.1016/j.apm.2018.09.021
19.
Ghayesh
,
M. H.
,
Farokhi
,
H.
, and
Alici
,
G.
,
2015
, “
Subcritical Parametric Dynamics of Microbeams
,”
Int. J. Eng. Sci.
,
95
, pp.
36
48
.10.1016/j.ijengsci.2015.06.001
20.
Farokhi
,
H.
, and
Ghayesh
,
M. H.
,
2018
, “
Supercritical Nonlinear Parameteric Dynamics of Timenshenko Microbeams
,”
Commun. Nonlinear Sci. Numer. Simul.
,
59
, pp.
592
605
.10.1016/j.cnsns.2017.11.033
21.
Wang
,
Y. G.
,
Lin
,
W. H.
,
Zhou
,
C. L.
, and
Liu
,
R. X.
,
2015
, “
Thermal Postbuckling and Free Vibration of Extensible Microscale Beams Based on Modified Couple Stress Theory
,”
J. Mech.
,
31
(
1
), pp.
37
46
.10.1017/jmech.2014.47
22.
Xia
,
W.
,
Wang
,
L.
, and
Yin
,
L.
,
2010
, “
Nonlinear Non-Classical Microscales Beams: Static Bending, Postbuckling and Free Vibration
,”
Int. J. Eng. Sci.
,
48
(
12
), pp.
2044
2053
.10.1016/j.ijengsci.2010.04.010
23.
Arda
,
M.
, and
Aydogdu
,
M.
,
2019
, “
Dynamic Stability of Harmonically Excited Nanobeams Including Axial Inertial
,”
J. Vib. Control
,
25
(
4
), pp.
820
833
.10.1177/1077546318802430
24.
Karimi
,
A. H.
, and
Rad
,
S. Z.
,
2015
, “
Nonlinear Coupled Longitudinal-Transverse Vibration Analysis of a Beam Subjected to a Moving Mass Traveling With Variable Speed
,”
Arch. Appl. Mech.
,
85
(
12
), pp.
1941
1960
.10.1007/s00419-015-1028-1
25.
Ghayesh
,
M. H.
, and
Farajpour
,
A.
,
2019
, “
A Review on the Mechanics of Functionally Graded Nanoscale and Microscale Structures
,”
Int. J. Eng. Sci.
,
137
, pp.
8
36
.10.1016/j.ijengsci.2018.12.001
26.
Zhu
,
X. W.
,
Wang
,
Y. B.
, and
Lou
,
Z. M.
,
2016
, “
A Study of the Critical Strain of Hyperelastic Materials: A New Kinematic Frame and the Leading Order Term
,”
Mech. Res. Commun.
,
78
, pp.
20
24
.10.1016/j.mechrescom.2016.10.007
27.
Bower
,
A. F.
,
2009
,
Applied Mechanics of Solids
, 1st ed.,
CRC Press
,
Boca Raton, FL
.
28.
Krack
,
M.
, and
Gross
,
J.
,
2019
,
Harmonic Balance for Nonlinear Vibration Problems
, 1st ed.,
Springer
,
Cham, Switzerland
.
29.
Allgower
,
E. L.
, and
Georg
,
K.
,
1990
,
Introduce to Numerical Continuation Methods
, 1st ed.,
Society for Industrial and Applied Mathematics
,
Philadephia, PA
.
You do not currently have access to this content.