This paper proposes an isolation transmissibility for the bending vibration of elastic beams. At both ends, the elastic beam is considered with vertical spring support and free to rotate. The geometric nonlinearity is considered. In order to implement the Galerkin method, the natural modes and frequencies of the bending vibration of the beam are analyzed. In addition, for the first time, the elastic continuum supported by boundary springs is solved by direct numerical method, such as the finite difference method (FDM). Moreover, the detailed procedure of FDM processing boundary conditions and initial conditions is presented. Two numerical approaches are compared to illustrate the correctness of the results. By demonstrating the significant impact, the necessity of elastic support at the boundaries to the vibration isolation of elastic continua is explained. Compared with the vibration transmission with one-term Galerkin truncation, it is proved that it is necessary to consider the high-order bending vibration modes when studying the force transmission of the elastic continua. Furthermore, the numerical examples illustrate that the influences of the system parameters on the bending vibration isolation. This study opens up the research on the vibration isolation of elastic continua, which is of profound significance to the analysis and design of vibration isolation for a wide range of practical engineering applications.

References

References
1.
Ibrahim
,
R. A.
,
2008
, “
Recent Advances in Nonlinear Passive Vibration Isolators
,”
J. Sound Vib.
,
314
(3–5), pp.
371
452
.
2.
Lang
,
Z. Q.
,
Jing
,
X. J.
,
Billings
,
S. A.
,
Tomlinson
,
G. R.
, and
Peng
,
Z. K.
,
2009
, “
Theoretical Study of the Effects of Nonlinear Viscous Damping on Vibration Isolation of SDOF Systems
,”
J. Sound Vib.
,
323
(1–2), pp.
352
365
.
3.
Laalej
,
H.
,
Lang
,
Z. Q.
,
Daley
,
S.
,
Zazas
,
I.
,
Billings
,
S. A.
, and
Tomlinson
,
G. R.
,
2012
, “
Application of Non-Linear Damping to Vibration Isolation: An Experimental Study
,”
Nonlinear Dyn.
,
69
(1–2), pp.
409
421
.
4.
Peng
,
Z. K.
,
Meng
,
G.
,
Lang
,
Z. Q.
,
Zhang
,
W. M.
, and
Chu
,
F. L.
,
2012
, “
Study of the Effects of Cubic Nonlinear Damping on Vibration Isolations Using Harmonic Balance Method
,”
Int. J. Non-Linear Mech.
,
47
(10), pp.
1073
1080
.
5.
Shen
,
Y. J.
,
Yang
,
S. P.
,
Xing
,
H. J.
, and
Ma
,
H. X.
,
2015
, “
Design of Single Degree-of-Freedom Optimally Passive Vibration Isolation System
,”
J. Vib. Eng. Technol.
,
3
(1), pp.
25
36
.http://www.tvi-in.com/Journals/journaldetail.aspx?Id=20150302181657848750099f6d3f311
6.
Fulcher
,
B. A.
,
Shahan
,
D. W.
,
Haberman
,
M. R.
,
Seepersad
,
C. C.
, and
Wilson
,
P. S.
,
2014
, “
Analytical and Experimental Investigation of Buckled Beams as Negative Stiffness Elements for Passive Vibration and Shock Isolation Systems
,”
ASME J. Vib. Acoust.
,
136
(3), p.
031009
.
7.
Huang
,
X. C.
,
Liu
,
X. T.
,
Sun
,
J. Y.
,
Zhang
,
Z. Y.
, and
Hua
,
H. X.
,
2014
, “
Vibration Isolation Characteristics of a Nonlinear Isolator Using Euler Buckled Beam as Negative Stiffness Corrector: A Theoretical and Experimental Study
,”
J. Sound Vib.
,
333
(4), pp.
1132
1148
.
8.
Lu
,
Z. Q.
,
Chen
,
L. Q.
,
Brennan
,
M. J.
,
Yang
,
T. J.
,
Ding
,
H.
, and
Liu
,
Z. G.
,
2016
, “
Stochastic Resonance in a Nonlinear Mechanical Vibration Isolation System
,”
J. Sound Vib.
,
370
, pp.
221
229
.
9.
Lu
,
Z. Q.
,
Brennan
,
M. J.
,
Yang
,
T. J.
,
Li
,
X. H.
, and
Liu
,
Z. G.
,
2013
, “
An Investigation of a Two-Stage Nonlinear Vibration Isolation System
,”
J. Sound Vib.
,
332
(6), pp.
1456
1464
.
10.
Wang
,
X. L.
,
Zhou
,
J. X.
,
Xu
,
D. L.
,
Ouyang
,
H. J.
, and
Duan
,
Y.
,
2017
, “
Force Transmissibility of a Two-Stage Vibration Isolation System With Quasi-Zero Stiffness
,”
Nonlinear Dyn.
,
87
(1), pp.
633
646
.
11.
Zhang
,
Y. W.
,
Fang
,
B.
, and
Zang
,
J.
,
2015
, “
Dynamic Features of Passive Whole-Spacecraft Vibration Isolation Platform Based on Non-Probabilistic Reliability
,”
J. Vib. Control
,
21
(1), pp.
60
67
.
12.
Niu
,
F.
,
Meng
,
L. S.
,
Wu
,
W. J.
,
Sun
,
J. G.
,
Su
,
W. H.
,
Meng
,
G.
, and
Rao
,
Z. S.
,
2013
, “
Recent Advances in Quasi-Zero-Stiffness Vibration Isolation Systems
,”
Appl. Mech. Mater.
,
397–400
, pp.
295
303
.
13.
Liu
,
X. L.
,
Shangguan
,
W. B.
,
Jing
,
X. J.
, and
Ahmed
,
W.
,
2016
, “
Vibration Isolation Analysis of Clutches Based on Trouble Shooting of Vehicle Accelerating Noise
,”
J. Sound Vib.
,
382
, pp.
84
99
.
14.
Xu
,
Z. D.
,
Xu
,
F. H.
, and
Chen
,
X.
,
2016
, “
Vibration Suppression on a Platform by Using Vibration Isolation and Mitigation Devices
,”
Nonlinear Dyn.
,
83
(3), pp.
1341
1353
.
15.
Zhang
,
J. R.
,
Guo
,
Z. X.
, and
Zhang
,
Y.
,
2016
, “
Dynamic Characteristics of Vibration Isolation Platforms Considering the Joints of the Struts
,”
Acta Astronaut.
,
126
, pp.
120
137
.
16.
Zhang
,
J. R.
,
Guo
,
Z. X.
,
Zhang
,
Y.
,
Tang
,
L.
, and
Guan
,
X.
,
2016
, “
Inner Structural Vibration Isolation Method for a Single Control Moment Gyroscope
,”
J. Sound Vib.
,
361
, pp.
78
98
.
17.
Liu
,
C. C.
,
Jing
,
X. J.
,
Daley
,
S.
, and
Li
,
F. M.
,
2015
, “
Recent Advances in Micro-Vibration Isolation
,”
Mech. Syst. Signal Process.
,
56–57
, pp.
55
80
.
18.
El-Khoury
,
O.
, and
Adeli
,
H.
,
2013
, “
Recent Advances on Vibration Control of Structures Under Dynamic Loading
,”
Arch. Comput. Methods Eng.
,
20
(4), pp.
353
360
.
19.
Elishakoff
,
I.
,
Lin
,
Y. K.
, and
Zhu
,
L. P.
,
1995
, “
Random Vibration of Uniform Beams With Varying Boundary-Conditions by the Dynamic-Edge-Effect Method
,”
Comput. Methods Appl. Mech. Eng.
,
121
(1–4), pp.
59
76
.
20.
Kim
,
H. K.
, and
Kim
,
M. S.
,
2001
, “
Vibration of Beams With Generally Restrained Boundary Conditions Using Fourier Series
,”
J. Sound Vib.
,
245
(5), pp.
771
784
.
21.
Kang
,
K. H.
, and
Kim
,
K. J.
,
1996
, “
Modal Properties of Beams and Plates on Resilient Supports With Rotational and Translational Complex Stiffness
,”
J. Sound Vib.
,
190
(2), pp.
207
220
.
22.
Chang
,
T. P.
, and
Chang
,
C. Y.
,
1998
, “
Vibration Analysis of Beams With a Two Degree-of-Freedom Spring-Mass System
,”
Int. J. Solids Struct.
,
35
(5–6), pp.
383
401
.
23.
Grossi
,
R. O.
, and
Quintana
,
M. V.
,
2008
, “
The Transition Conditions in the Dynamics of Elastically Restrained Beams
,”
J. Sound Vib.
,
316
(1–5), pp.
274
297
.
24.
Xing
,
J. Z.
, and
Wang
,
Y. G.
,
2013
, “
Free Vibrations of a Beam With Elastic End Restraints Subject to a Constant Axial Load
,”
Arch. Appl. Mech.
,
83
(2), pp.
241
252
.
25.
Lee
,
S. Y.
, and
Lin
,
S. M.
,
1998
, “
Non-Uniform Timoshenko Beams With Time-Dependent Elastic Boundary Conditions
,”
J. Sound Vib.
,
217
(2), pp.
223
238
.
26.
Zhang
,
S.
, and
Cheng
,
L.
,
2016
, “
On the Efficacy of the Wavelet Decomposition for High Frequency Vibration Analyses
,”
J. Sound Vib.
,
380
, pp.
213
223
.
27.
Jin
,
J. D.
,
Yang
,
X. D.
, and
Zhang
,
Y. F.
,
2011
, “
Stability and Natural Characteristics of a Supported Beam
,”
Adv. Mater. Res.
,
338
, pp.
467
472
.
28.
Wang
,
Y. R.
, and
Fang
,
Z. W.
,
2015
, “
Vibrations in an Elastic Beam With Nonlinear Supports at Both Ends
,”
J. Appl. Mech. Tech. Phys.
,
56
(2), pp.
337
346
.
29.
Wang
,
Y. Q.
,
Liang
,
L.
, and
Guo
,
X. H.
,
2013
, “
Internal Resonance of Axially Moving Laminated Circular Cylindrical Shells
,”
J. Sound Vib.
,
332
(24), pp.
6434
6450
.
30.
Ding
,
H.
, and
Zu
,
J. W.
,
2014
, “
Steady-State Responses of Pulley-Belt Systems With a One-Way Clutch and Belt Bending Stiffness
,”
ASME J. Vib. Acoust.
,
136
(4), p.
041006
.
31.
Wattanasakulpong
,
N.
, and
Chaikittiratana
,
A.
,
2014
, “
On the Linear and Nonlinear Vibration Responses of Elastically End Restrained Beams Using DTM
,”
Mech. Based Des. Struct. Mach.
,
42
(2), pp.
135
150
.
32.
Wattanasakulpong
,
N.
, and
Chaikittiratana
,
A.
,
2016
, “
Adomian-Modified Decomposition Method for Large-Amplitude Vibration Analysis of Stepped Beams With Elastic Boundary Conditions
,”
Mech. Based Des. Struct. Mach.
,
44
(3), pp.
270
282
.
33.
Yang
,
Y. F.
,
Feng
,
H. B.
, and
Chen
,
H.
,
2016
, “
Dynamical Analysis of the Flexible Beam-Cam Oblique-Impact System
,”
Acta Phys. Sin.
,
65
(24), p.
240502
.http://wulixb.iphy.ac.cn/EN/Y2016/V65/I24/240502
34.
Ding
,
H.
,
Zhang
,
G. C.
,
Chen
,
L. Q.
, and
Yang
,
S. P.
,
2012
, “
Forced Vibrations of Supercritically Transporting Viscoelastic Beams
,”
ASME J. Vib. Acoust.
,
134
(5), p.
051007
.
You do not currently have access to this content.