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Research Papers

Stochastic Averaging for Quasi-Integrable Hamiltonian Systems With Variable Mass

[+] Author and Article Information
Yong Wang

Department of Engineering Mechanics,
Zhejiang University,
#38 Zheda Road,
Hangzhou, Zhejiang 310027, China
e-mail: yongpi.wang@gmail.com

Xiaoling Jin

Department of Engineering Mechanics,
Zhejiang University,
#38 Zheda Road,
Hangzhou, Zhejiang 310027, China
e-mail: jinling113@gmail.com

Zhilong Huang

Department of Engineering Mechanics,
Zhejiang University,
#38 Zheda Road,
Hangzhou, Zhejiang 310027, China
e-mail: zlhuang@zju.edu.cn

1Corresponding author.

Manuscript received August 3, 2013; final manuscript received November 5, 2013; accepted manuscript posted November 11, 2013; published online December 10, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(5), 051003 (Dec 10, 2013) (7 pages) Paper No: JAM-13-1325; doi: 10.1115/1.4025954 History: Received August 03, 2013; Revised November 05, 2013; Accepted November 11, 2013

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.

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References

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. [CrossRef]
Zhu, W. Q., 1996, “Recent Developments and Applications of Stochastic Averaging Method in Random Vibration,” ASME Appl. Mech. Rev., 49(10), pp. 572–580. [CrossRef]
Stratonovich, R. L., 1963, Topics in the Theory of Random Noise, Gordon and Breach, New York.
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. [CrossRef]
Zhu, W. Q., and Lin, Y. K., 1991, “Stochastic Averaging of Energy Envelope,” ASCE J. Eng. Mech., 117, pp. 1890–1905. [CrossRef]
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. [CrossRef]
Zhu, W. Q., and Yang, Y. Q., 1997, “Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems,” ASME J. Appl. Mech., 64, pp. 157–164. [CrossRef]
Zhu, W. Q., 2006, “Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation,” ASME Appl. Mech. Rev., 59, pp. 230–248. [CrossRef]
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. [CrossRef]
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. [CrossRef] [PubMed]
NuthIII, 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. [CrossRef] [PubMed]
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. [CrossRef]
Banerjee, A. K., 2000, “Dynamics of a Variable-Mass, Flexible-Body System,” J. Guid. Control Dyn., 23(3), pp. 501–508. [CrossRef]
Wang, S. M., and Eke, F. O., 1995, “Rotational Dynamics of Axisymmetric Variable Mass Systems,” ASME J. Appl. Mech., 62, pp. 970–974. [CrossRef]
Cveticanin, L., and Kovacic, I., 2007, “On the Dynamics of Bodies With Continual Mass Variation,” ASME J. Appl. Mech., 74, pp. 810–815. [CrossRef]
Kendoush, A. A., 2005, “The Virtual Mass of a Rotating Sphere in Fluids,” ASME J. Appl. Mech., 72, pp. 801–802. [CrossRef]
van Brummelen, E. H., 2009, “Added Mass Effects of Compressible and Incompressible Flows in Fluid-Structure Interaction,” ASME J. Appl. Mech., 76, p. 021206. [CrossRef]
Indeitsev, D. A., and Semenov, B. N., 2008, “About One Model of Structural-Phase Transformations Under Hydrogen Influence,” Acta Mech., 195, pp. 295–304. [CrossRef]
Cveticanin, L., 1998, Dynamics of Machines with Variable Mass, Gordon and Breach, New York.
Flores, J., Solovey, G., and Gil, S., 2003, “Variable Mass Oscillator,” Am. J. Phys., 71(7), pp. 721–725. [CrossRef]
Cveticanin, L., 2012, “Oscillator With Non-Integer Order Nonlinearity and Time Variable Parameters,” Acta Mech., 223, pp. 1417–1429. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Bashir, R., 2004, “BioMEMS: State-of-the-Art in Detection, Opportunities and Prospects,” Adv. Drug Deliv. Rev., 56, pp. 1565–1586. [CrossRef] [PubMed]
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. [CrossRef]
Boisen, A., Dohn, S., Keller, S. S., Schmid, S., and Tenje, M., 2011, “Cantilever-Like Micromechanical Sensors,” Rep. Prog. Phys., 74, p. 036101. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef]
Cornelisse, J. W., Schoyer, H. F. R., and Wakker, K. F., 1979, Rocket Propulsion and Spaceflight Dynamics, Pitman, London.
Chen, B., 2012, Analytical Dynamics, Peking University, Beijing (in Chinese).
Lin, Y. K., and Cai, G. Q., 1995, Probabilistic Structural Dynamics: Advanced Theory and Application, McGraw-Hill, New York.
Khasminskii, R. Z., 1968, “On the Averaging Principle for Itô Stochastic Differential Equations,” Kybernetika, 3, pp. 260–279 (in Russian).
Bendat, J. S., and Piersol, A. G., 2000, Random Data: Analysis and Measurement Procedures, Wiley, New York.
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. [CrossRef]
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. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0 on mass disturbance intensity 2Dm. (2De=0.2, b=0.2, c=0.01). —, analytical results; •, ♦, results from MCS.

Grahic Jump Location
Fig. 2

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0on external excitation intensity 2De. (2Dm=0.1, b=0.2, c=0.01). —, analytical results; •, ♦, results from MCS.

Grahic Jump Location
Fig. 3

Stationary probability densities of generalized displacement Q1 and generalized momentum P1 for three cases: proposed analytical results, MCS for original system in Eq. (28), and MCS for approximate system in Eq. (29). (2Dm=0.1, 2De=0.2, b=0.2, c=0.01).

Grahic Jump Location
Fig. 4

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0 on damping coefficient c. (2Dm=0.1, 2De=0.2, b=0.2).

Grahic Jump Location
Fig. 5

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0 on nonlinear stiffness coefficient b. (2Dm=0.1, 2De=0.2, c=0.01).

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