Variable-mass systems become more and more important with the explosive development of micro- and nanotechnologies, and it is crucial to evaluate the influence of mass disturbances on system random responses. This manuscript generalizes the stochastic averaging technique from quasi-integrable Hamiltonian systems to stochastic variable-mass systems. The Hamiltonian equations for variable-mass systems are firstly derived in classical mechanics formulation and are approximately replaced by the associated conservative Hamiltonian equations with disturbances in each equation. The averaged Itô equations with respect to the integrals of motion as slowly variable processes are derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the joint probability densities of the integrals of motion. A representative variable-mass oscillator is worked out to demonstrate the application and effectiveness of the generalized stochastic averaging technique; also, the sensitivity of random responses to pivotal system parameters is illustrated.

References

References
1.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1986
, “
Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems
,”
Int. J. Non-Linear Mech.
,
21
, pp.
111
134
.10.1016/0020-7462(86)90025-9
2.
Zhu
,
W. Q.
,
1996
, “
Recent Developments and Applications of Stochastic Averaging Method in Random Vibration
,”
ASME Appl. Mech. Rev.
,
49
(
10
), pp.
572
580
.10.1115/1.3101980
3.
Stratonovich
,
R. L.
,
1963
,
Topics in the Theory of Random Noise
,
Gordon and Breach
,
New York
.
4.
Khasminskii
,
R. Z.
,
1966
, “
A Limit Theorem for Solutions of Differential Equations With Random Right Hand Sides
,”
Theory of Probab. Appl.
,
11
, pp.
390
405
.10.1137/1111038
5.
Zhu
,
W. Q.
, and
Lin
,
Y. K.
,
1991
, “
Stochastic Averaging of Energy Envelope
,”
ASCE J. Eng. Mech.
,
117
, pp.
1890
1905
.10.1061/(ASCE)0733-9399(1991)117:8(1890)
6.
Zhu
,
W. Q.
,
Huang
,
Z. L.
, and
Yang
,
Y. Q.
,
1997
, “
Stochastic Averaging of Quasi-Integrable Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
64
, pp.
975
984
.10.1115/1.2789009
7.
Zhu
,
W. Q.
, and
Yang
,
Y. Q.
,
1997
, “
Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
64
, pp.
157
164
.10.1115/1.2787267
8.
Zhu
,
W. Q.
,
2006
, “
Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation
,”
ASME Appl. Mech. Rev.
,
59
, pp.
230
248
.10.1115/1.2193137
9.
Lopez
,
G.
,
Barrera
,
L. A.
,
Garibo
,
Y.
,
Hernandez
,
H.
,
Salazar
,
J. C.
, and
Vargas
,
C. A.
,
2004
, “
Constants of Motion for Several One-Dimensional Systems and Problems Associated With Getting Their Hamiltonians
,”
Int. J. Theor. Phys.
,
43
(
10
), pp.
2009
2021
.10.1023/B:IJTP.0000049006.61937.c5
10.
Zagorodny
,
A. G.
,
Schram
,
P. P. J. M.
, and
Trigger
,
S. A.
,
2000
, “
Stationary Velocity and Charge Distributions of Grains in Dusty Plasmas
,”
Phys. Rev. Lett.
,
84
(
16
), pp.
3594
3597
.10.1103/PhysRevLett.84.3594
11.
Nuth
III,
J. A.
,
Hill
,
H. G. M.
, and
Kletetschka
,
G.
,
2000
, “
Determining the Ages of Comets From the Fraction of Crystalline Dust
,”
Nature
,
406
, pp.
275
276
.10.1038/35018516
12.
Lopez
,
G.
,
2007
, “
Constant of Motion, Lagrangian and Hamiltonian of the Gravitational Attraction of Two Bodies With Variable Mass
,”
Int. J. Theor. Phys.
,
46
(
4
), pp.
806
816
.10.1007/s10773-006-9085-4
13.
Banerjee
,
A. K.
,
2000
, “
Dynamics of a Variable-Mass, Flexible-Body System
,”
J. Guid. Control Dyn.
,
23
(
3
), pp.
501
508
.10.2514/2.4556
14.
Wang
,
S. M.
, and
Eke
,
F. O.
,
1995
, “
Rotational Dynamics of Axisymmetric Variable Mass Systems
,”
ASME J. Appl. Mech.
,
62
, pp.
970
974
.10.1115/1.2896031
15.
Cveticanin
,
L.
, and
Kovacic
,
I.
,
2007
, “
On the Dynamics of Bodies With Continual Mass Variation
,”
ASME J. Appl. Mech.
,
74
, pp.
810
815
.10.1115/1.2711231
16.
Kendoush
,
A. A.
,
2005
, “
The Virtual Mass of a Rotating Sphere in Fluids
,”
ASME J. Appl. Mech.
,
72
, pp.
801
802
.10.1115/1.1989357
17.
van Brummelen
,
E. H.
,
2009
, “
Added Mass Effects of Compressible and Incompressible Flows in Fluid-Structure Interaction
,”
ASME J. Appl. Mech.
,
76
, p.
021206
.10.1115/1.3059565
18.
Indeitsev
,
D. A.
, and
Semenov
,
B. N.
,
2008
, “
About One Model of Structural-Phase Transformations Under Hydrogen Influence
,”
Acta Mech.
,
195
, pp.
295
304
.10.1007/s00707-007-0568-z
19.
Cveticanin
,
L.
,
1998
,
Dynamics of Machines with Variable Mass
,
Gordon and Breach
,
New York
.
20.
Flores
,
J.
,
Solovey
,
G.
, and
Gil
,
S.
,
2003
, “
Variable Mass Oscillator
,”
Am. J. Phys.
,
71
(
7
), pp.
721
725
.10.1119/1.1571838
21.
Cveticanin
,
L.
,
2012
, “
Oscillator With Non-Integer Order Nonlinearity and Time Variable Parameters
,”
Acta Mech.
,
223
, pp.
1417
1429
.10.1007/s00707-012-0665-5
22.
Fukuma
,
T.
,
Kimura
,
M.
,
Kobayashi
,
K.
,
Matsushige
,
K.
, and
Yamada
,
H.
,
2005
, “
Development of Low Noise Cantilever Deflection Sensor for Multi-Environment Frequency-Modulation Atomic Force Microscopy
,”
Rev. Sci. Instrum.
,
76
, p.
053704
.10.1063/1.1896938
23.
Shaw
,
S. W.
, and
Balachandren
,
B.
,
2008
, “
A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008
,”
J. Syst. Des. Dyn.
,
2
(
3
), pp.
611
640
.10.1299/jsdd.2.611
24.
Bashir
,
R.
,
2004
, “
BioMEMS: State-of-the-Art in Detection, Opportunities and Prospects
,”
Adv. Drug Deliv. Rev.
,
56
, pp.
1565
1586
.10.1016/j.addr.2004.03.002
25.
Lavrik
,
N. V.
,
Sepaniak
,
M. J.
, and
Datskos
,
P. G.
,
2004
, “
Cantilever Transducers as a Platform for Chemical and Biological Sensors
,”
Rev. Sci. Instrum.
,
75
, pp.
2229
2253
.10.1063/1.1763252
26.
Boisen
,
A.
,
Dohn
,
S.
,
Keller
,
S. S.
,
Schmid
,
S.
, and
Tenje
,
M.
,
2011
, “
Cantilever-Like Micromechanical Sensors
,”
Rep. Prog. Phys.
,
74
, p.
036101
.10.1088/0034-4885/74/3/036101
27.
Tamayo
,
J.
,
Kosaka
,
P. M.
,
Ruz
,
J. J.
,
Paulo
,
A. S.
, and
Calleja
,
M.
,
2013
, “
Biosensors Based on Nanomechanical Systems
,”
Chem. Soc. Rev.
,
42
, pp.
1287
1311
.10.1039/c2cs35293a
28.
Lee
,
P. S.
,
Lee
,
J.
,
Shin
,
N.
,
Lee
,
K. H.
,
Lee
,
D.
,
Jeon
,
S.
,
Choi
,
D.
,
Hwang
,
W.
, and
Park
,
H.
,
2008
, “
Microcantilevers With Nanochannels
,”
Adv. Mater.
,
20
(
9
), pp.
1732
1737
.10.1002/adma.200701490
29.
Datar
,
R.
,
Kim
,
S.
,
Jeon
,
S.
,
Hesketh
,
P.
,
Manalis
,
S.
,
Boisen
,
A.
, and
Thundat
,
T.
,
2009
, “
Cantilever Sensors: Nanomechanical Tools for Diagnostics
,”
MRS Bull.
,
34
, pp.
449
454
.10.1557/mrs2009.121
30.
Cornelisse
,
J. W.
,
Schoyer
,
H. F. R.
, and
Wakker
,
K. F.
,
1979
,
Rocket Propulsion and Spaceflight Dynamics
,
Pitman
,
London
.
31.
Chen
,
B.
,
2012
,
Analytical Dynamics
,
Peking University
,
Beijing
(in Chinese).
32.
Lin
,
Y. K.
, and
Cai
,
G. Q.
,
1995
,
Probabilistic Structural Dynamics: Advanced Theory and Application
,
McGraw-Hill
,
New York
.
33.
Khasminskii
,
R. Z.
,
1968
, “
On the Averaging Principle for Itô Stochastic Differential Equations
,”
Kybernetika
,
3
, pp.
260
279
(in Russian).
34.
Bendat
,
J. S.
, and
Piersol
,
A. G.
,
2000
,
Random Data: Analysis and Measurement Procedures
,
Wiley
,
New York
.
35.
Zeng
,
Y.
, and
Zhu
,
W. Q.
,
2011
, “
Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems Under Poisson White Noise Excitation
,”
ASME J. Appl. Mech.
,
78
, p.
021002
.10.1115/1.4002528
36.
Jia
,
W. T.
,
Zhu
,
W. Q.
,
Xu
,
Y.
, and
Liu
,
W. Y.
,
2013
, “
Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations
,”
ASME J. Appl. Mech.
,
81
(
4
), p.
041009
.10.1115/1.4025141
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