Abstract

The nonlinear vibrations of elastic beams with large amplitudes are frequently treated as a typical problem of an elastica. As the continuation of the analysis of the deformation of an elastica, the nonlinear vibration equation of the elastic beam in the rotation angle of the cross section has been established. Using the deformation function, the nonlinear equation with the inertia effect has been solved by the newly proposed extended Galerkin method (EGM). The solution to the vibration problem of the elastica is compared with earlier approximate solutions including the frequencies and mode shapes obtained by other methods, and the rotation angle and energy of each mode at the high-order frequency are also calculated. This solution procedure provides an alternative technique to the elastica problem by the EGM with possible applications to other nonlinear problems in many fields of science and technology.

References

1.
Bahrami
,
M. N.
,
Arani
,
M. K.
, and
Saleh
,
N. R.
,
2011
, “
Modified Wave Approach for the Calculation of Natural Frequencies and Mode Shapes in Arbitrary Non-Uniform Beams
,”
Scientia Iranica
,
18
(
5
), pp.
1088
1094
.10.1016/j.scient.2011.08.004
2.
Freno
,
B. A.
, and
Cizmas
,
P. G. A.
,
2011
, “
A Computationally Efficient Non-Linear Beam Model
,”
Int. J. Non-Linear Mech.
,
46
(
6
), pp.
854
869
.10.1016/j.ijnonlinmec.2011.03.010
3.
Sinha
,
A.
,
2022
, “
Computing Natural Frequencies and Mode Shapes of an Axially Moving Nonuniform Beam
,”
ASME. J. Comput. Nonlinear Dynam.
,
17
(
4
), p. 041001.10.1115/1.4053271
4.
Wang
,
Y.
, and
Zhu
,
W.
,
2021
, “
Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading
,”
ASME. J. Comput. Nonlinear Dynam.
,
16
(
3
), p. 031006.10.1115/1.4049562
5.
Utzeri
,
M.
,
Sasso
,
M.
,
Chiappini
,
G.
, and
Lenci
,
S.
,
2020
, “
Nonlinear Vibrations of a Composite Beam in Large Displacements: Analytical, Numerical, and Experimental Approaches
,”
ASME. J. Comput. Nonlinear Dynam.
,
16
(
2
), p. 021002.10.1115/1.4048913
6.
Nikolić
,
A.
, and
Šalinić
,
S.
,
2020
, “
Free Vibration Analysis of Cracked Beams by Using Rigid Segment Method
,”
Appl. Math. Modell.
,
84
, pp.
158
172
.10.1016/j.apm.2020.03.033
7.
Repetto
,
C. E.
,
Roatta
,
A.
, and
Welti
,
R. J.
,
2012
, “
Forced Vibrations of a Cantilever Beam
,”
Eur. J. Phys.
,
33
(
5
), pp.
1187
1195
.10.1088/0143-0807/33/5/1187
8.
Jaworski
,
J. W.
, and
Dowell
,
E. H.
,
2008
, “
Free Vibration of a Cantilevered Beam With Multiple Steps: Comparison of Several Theoretical Methods With Experiment
,”
J. Sound Vib.
,
312
(
4–5
), pp.
713
725
.10.1016/j.jsv.2007.11.010
9.
Sedighi
,
H. M.
,
Shirazi
,
K. H.
, and
Noghrehabadi
,
A.
,
2012
, “
Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
13
(
7–8
), pp.
487
494
.10.1515/ijnsns-2012-0030
10.
Shang-Rou
,
H.
,
Shaw
,
S. W.
, and
Pierre
,
C.
,
1994
, “
Normal Modes for Large Amplitude Vibration of a Cantilever Beam
,”
Int. J. Solids Struct.
,
31
(
14
), pp.
1981
2014
.10.1016/0020-7683(94)90203-8
11.
Gritsenko
,
D.
,
Xu
,
J.
, and
Paoli
,
R.
,
2020
, “
Transverse Vibrations of Cantilever Beams: Analytical Solutions With General Steady-State Forcing
,”
Appl. Eng. Sci.
,
3
, p.
100017
.10.1016/j.apples.2020.100017
12.
Yaman
,
M.
, and
Sen
,
S.
,
2007
, “
Vibration Control of a Cantilever Beam of Varying Orientation
,”
Int. J. Solids Struct.
,
44
(
3–4
), pp.
1210
1220
.10.1016/j.ijsolstr.2006.06.015
13.
Ullah
,
M. W.
,
Rahman
,
M. S.
, and
Uddin
,
M. A.
,
2022
, “
Free Vibration Analysis of Nonlinear Axially Loaded Beams Using a Modified Harmonic Balance Method
,”
Partial Differ. Equ. Appl. Math.
,
6
, p.
100414
.10.1016/j.padiff.2022.100414
14.
Barari
,
A.
,
Kaliji
,
H. D.
,
Ghadimi
,
M.
, and
Domairry
,
G.
,
2011
, “
Non-Linear Vibration of Euler-Bernoulli Beams
,”
Latin Am. J. Solids Struct.
,
8
(
2
), pp.
139
148
.10.1590/S1679-78252011000200002
15.
Xie
,
L.
,
Wang
,
S.
,
Ding
,
J.
,
Banerjee
,
J. R.
, and
Wang
,
J.
,
2020
, “
An Accurate Beam Theory and Its First-Order Approximation in Free Vibration Analysis
,”
J. Sound Vib.
,
485
, p.
115567
.10.1016/j.jsv.2020.115567
16.
Jing
,
H.
,
Gong
,
X.
,
Wang
,
J.
,
Wu
,
R.
, and
Huang
,
B.
,
2022
, “
An Analysis of Nonlinear Beam Vibrations With the Extended Rayleigh-Ritz Method
,”
J. Appl. Comput. Mech.
,
8
(
4
), pp.
1299
1306
.10.22055/JACM.2022.39580.3434
17.
Liu
,
Y.
, and
Gurram
,
C. S.
,
2009
, “
The Use of He's Variational Iteration Method for Obtaining the Free Vibration of an Euler–Bernoulli Beam
,”
Math. Comput. Modell.
,
50
(
11–12
), pp.
1545
1552
.10.1016/j.mcm.2009.09.005
18.
Chen
,
Y.
,
Zhang
,
J.
, and
Zhang
,
H.
,
2017
, “
Free Vibration Analysis of Rotating Tapered Timoshenko Beams Via Variational Iteration Method
,”
J. Vib. Control
,
23
(
2
), pp.
220
234
.10.1177/1077546315576431
19.
Ozturk
,
B.
,
2009
, “
Free Vibration Analysis of Beam on Elastic Foundation by the Variational Iteration Method
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
10
(
10
), pp.
1255
1262
.10.1515/IJNSNS.2009.10.10.1255
20.
Rezaee
,
M.
, and
Hassannejad
,
R.
,
2011
, “
A New Approach to Free Vibration Analysis of a Beam With a Breathing Crack Based on Mechanical Energy Balance Method
,”
Acta Mech. Solida Sin.
,
24
(
2
), pp.
185
194
.10.1016/S0894-9166(11)60020-7
21.
Mehdipour
,
I.
,
Ganji
,
D. D.
, and
Mozaffari
,
M.
,
2010
, “
Application of the Energy Balance Method to Nonlinear Vibrating Equations
,”
Curr. Appl. Phys.
,
10
(
1
), pp.
104
112
.10.1016/j.cap.2009.05.016
22.
Bayat
,
M.
, and
Pakar
,
I.
,
2011
, “
Application of He's Energy Balance Method for Nonlinear Vibration of Thin Circular Sector Cylinder
,”
Int. J. Phys. Sci.
,
6
(
23
), pp.
5564
5570
.10.5897/IJPS11.756
23.
Pirbodaghi
,
T.
,
Ahmadian
,
M. T.
, and
Fesanghary
,
M.
,
2009
, “
On the Homotopy Analysis Method for Non-Linear Vibration of Beams
,”
Mech. Res. Commun.
,
36
(
2
), pp.
143
148
.10.1016/j.mechrescom.2008.08.001
24.
Wang
,
J.
,
Chen
,
J. K.
, and
Liao
,
S. J.
,
2008
, “
An Explicit Solution of the Large Deformation of a Cantilever Beam Under Point Load at the Free Tip
,”
J. Comput. Appl. Math.
,
212
(
2
), pp.
320
330
.10.1016/j.cam.2006.12.009
25.
Samadani
,
F.
,
Moradweysi
,
P.
,
Ansari
,
R.
,
Hosseini
,
K.
, and
Darvizeh
,
A.
,
2017
, “
Application of Homotopy Analysis Method for the Pull-in and Nonlinear Vibration Analysis of Nanobeams Using a Nonlocal Euler–Bernoulli Beam Model
,”
Z. Für Naturforsch. A
,
72
(
12
), pp.
1093
1104
.10.1515/zna-2017-0174
26.
Motallebi
,
A. A.
,
Poorjamshidian
,
M.
, and
Sheikhi
,
J.
,
2014
, “
Vibration Analysis of a Nonlinear Beam Under Axial Force by Homotopy Analysis Method
,”
J. Solid Mech.
,
6
(
3
), pp.
289
298
.https://www.researchgate.net/publication/287957389_Vibration_analysis_of_a_nonlinear_beam_under_axial_force_by_Homotopy_analysis_method
27.
Sedighi
,
H. M.
, and
Reza
,
A.
,
2013
, “
High Precise Analysis of Lateral Vibration of Quintic Nonlinear Beam
,”
Latin Am. J. Solids Struct.
,
10
(
2
), pp.
441
452
.10.1590/S1679-78252013000200010
28.
Elishakoff
,
I.
, and
Zingales
,
M.
,
2004
, “
Convergence of Boobnov-Galerkin Method Exemplified
,”
AIAA J.
,
42
(
9
), pp.
1931
1933
.10.2514/1.898
29.
Elishakoff
,
I.
,
Amato
,
M.
, and
Marzani
,
A.
,
2021
, “
Galerkin's Method Revisited and Corrected in the Problem of Jaworski and Dowell
,”
Mech. Syst. Signal Process.
,
155
, p.
107604
.10.1016/j.ymssp.2020.107604
30.
Duncan
,
W. J.
,
1938
, “
LIV. Note on Galerkin's Method for the Treatment of Problems Concerning Elastic Bodies
,”
Philos. Mag. J. Sci.
,
25
(
169
), pp.
628
633
.10.1080/14786443808562046
31.
Elishakoff
,
I.
, and
Zingales
,
M.
,
2003
, “
Coincidence of Boobnov-Galerkin and Closed-Form Solutions in an Applied Mechanics Problem
,”
ASME J. Appl. Mech.
,
70
(
5
), pp.
777
779
.10.1115/1.1598474
32.
Zhang
,
J.
,
Wu
,
R.
,
Wang
,
J.
,
Ma
,
T.
, and
Wang
,
L.
,
2022
, “
The Approximate Solution of Nonlinear Flexure of a Cantilever Beam With the Galerkin Method
,”
Appl. Sci.
,
12
(
13
), p.
6720
.10.3390/app12136720
33.
Wang
,
J.
, and
Wu
,
R.
,
2022
, “
The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations
,”
Appl. Sci.
,
12
(
6
), p.
2979
.10.3390/app12062979
34.
Wu
,
H.
,
Wu
,
R.
,
Ma
,
T.
,
Lu
,
Z.
,
Li
,
H.
, and
Wang
,
J.
,
2022
, “
A Nonlinear Analysis of Surface Acoustic Waves in Isotropic Elastic Solids
,”
Theor. Appl. Mech. Lett.
,
12
(
2
), p.
100326
.10.1016/j.taml.2022.100326
35.
Shi
,
B.
,
Yang
,
J.
, and
Wang
,
J.
,
2021
, “
Forced Vibration Analysis of Multi-Degree-of-Freedom Nonlinear Systems With the Extended Galerkin Method
,”
Mech. Adv. Mater. Struct.
,
30
(
4
), pp.
794
802
.10.1080/15376494.2021.2023922
36.
Lian
,
C.
,
Meng
,
B.
,
Jing
,
H.
,
Wu
,
R.
,
Lin
,
J.
, and
Wang
,
J.
,
2023
, “
The Analysis of Higher Order Nonlinear Vibrations of an Elastic Beam With the Extended Galerkin Method
,”
J. Vib. Eng. Technol
.10.1007/s42417-023-01011-6
37.
Weaver
,
W. J.
,
Timoshenko
,
S. P.
, and
Young
,
D. H.
,
1991
,
Vibration Problems in Engineering
,
Wiley
,
New York,
Chap.
5
.
38.
Kopmaz
,
O.
, and
Gündoğdu
,
Ö.
,
2003
, “
On the Curvature of an Euler–Bernoulli Beam
,”
Int. J. Mech. Eng. Educ.
,
31
(
2
), pp.
132
142
.10.7227/IJMEE.31.2.5
39.
Asmar
,
N.
,
2016
,
Partial Differential Equations With Fourier Series and Boundary Value Problems
,
Courier Dover Publications
,
New York
.
40.
Evensen
,
D. A.
,
1968
, “
Nonlinear Vibrations of Beams With Various Boundary Conditions
,”
AIAA J.
,
6
(
2
), pp.
370
372
.10.2514/3.4506
41.
Huang
,
Q.
,
Wu
,
R.
,
Xie
,
L.
,
Zhang
,
A.
,
Huang
,
B.
,
Du
,
J.
, and
Wang
,
J.
,
2020
, “
Examinations of Vibration Frequency and Mode Shape Variations of Quartz Crystal Plates in a Thermal Field With Strain and Kinetic Energies
,”
J. Therm. Stresses
,
43
(
4
), pp.
456
472
.10.1080/01495739.2020.1722049
42.
Huang
,
Q.
,
Wu
,
R.
,
Wang
,
L.
,
Xie
,
L.
,
Du
,
J.
,
Ma
,
T.
, and
Wang
,
J.
,
2020
, “
Identification of Vibration Modes of Quartz Crystal Plates With Proportion of Strain and Kinetic Energies
,”
Int. J. Acoust. Vib.
,
25
(
3
), pp.
392
407
.10.20855/ijav.2020.25.31671
43.
Pavić
,
G.
,
2006
, “
Vibration Damping, Energy and Energy Flow in Rods and Beams: Governing Formulae and Semi-Infinite Systems
,”
J. Sound Vib.
,
291
(
3–5
), pp.
932
962
.10.1016/j.jsv.2005.07.021
You do not currently have access to this content.