An Improved Series Expansion of the Solution to the Moving Oscillator Problem

A. V. Pesterev; L. A. Bergman

doi:10.1115/1.568436

Journal of Vibration and Acoustics | Volume 122 | Issue 1 | TECHNICAL PAPERS

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TECHNICAL PAPERS

An Improved Series Expansion of the Solution to the Moving Oscillator Problem

A. V. Pesterev and L. A. Bergman

[+] Author and Article Information

A. V. Pesterev

Institute for Systems Analysis, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 9, Moscow, 119034 Russia

L. A. Bergman

Aeronautical and Astronautical Engineering Department, University of Illinois, Urbana, Illinois 61801

J. Vib. Acoust 122(1), 54-61 (Jan 01, 1999) (8 pages) doi:10.1115/1.568436 History: Received January 01, 1999

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References

Pesterev, A. V., and Bergman, L. A., 1997a, “Vibration of Elastic Continuum Carrying Moving Linear Oscillator,” ASCE J. Eng. Mech., 123, pp. 878–884.

Pesterev, A. V., and Bergman, L. A., 1997b, “Vibration of Elastic Continuum Carrying Accelerating Oscillator,” ASCE J. Eng. Mech., 123, pp. 886–889.

Pesterev, A. V., and Bergman, L. A., 1998a, “Response of a Nonconservative Continuous System to a Moving Concentrated Load,” ASME J. Appl. Mech., 65, pp. 436–444.

Pesterev, A. V., and Bergman, L. A., 1998b, “A Contribution to the Moving Mass Problem,” ASME J. Vibr. Acoust., 120, pp. 824–826.

Yang. B., Tan. C. A., and Bergman, L. A., 1998, “On the Problem of a Distributed Parameter System Carrying a Moving Oscillator,” Dynamics and Control of Distributed Systems, H. Tzou and L. Bergman, eds., Cambridge University Press.

Dowell, E. H., 1996, “Comment on Energy Flow Predictions in a Structure of Rigidly Joined Beams Using Receptance Theory,” J. Sound Vib., 194, pp. 445–447.

Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., 1955, Acroelasticity, Addison-Wesley Publishing Co., Boston: reprinted 1996, Dover, New York.

Yang, B., 1996, “Integral Formulas for Non-Self-Adjoint Distributed Dynamic Systems,” AIAA J., 34, pp. 2132–2139.

Sadiku, S., and Leipholz, H. H. E., 1987, “On the Dynamic of Elastic Systems with Moving Concentrated Masses,” Ingenieur Arch., 57, pp. 223–242.

Lamb, H., 1931, The Dynamical Theory of Sound, Edward Arnold & Co., London, 2nd edition.

Figures

Displacement of the moving mass on the CC beam for v=2 m/s: (i) exact solution (solid line), (ii) moving force approximation (dashed line), and (iii) quasi-static approximation (dotted line).

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Displacement of the moving mass on the CC beam for v=4 m/s: (i) exact solution (solid line), (ii) moving force approximation (dashed line), and (iii) quasi-static approximation (dotted line).

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Displacement of the moving mass on the CC beam for v=24 m/s: (i) exact solution (solid line), (ii) moving force approximation (dashed line), and (iii) quasi-static approximation (dotted line).

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The moving oscillator solution (solid line) for the CC beam and its moving force (dashed line) and quasi-static (dotted line) approximations

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Shape of the string w(x,t) at t=0.5 for the moving oscillator problem (solid line) and its moving force (dashed line) and quasi-static (dotted line) approximations

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Exact solution w(x,0.5) to the moving oscillator problem (solid line) (obtained by using 100 terms of the improved series) and two approximations to it obtained by means of five first terms of the improved series (dashed line) and five terms of the ordinary series (dotted line)

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Approximations of the moving oscillator solution for the SS beam by the improved series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

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Approximations of the moving oscillator solution for the SS beam by the ordinary series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

View Large | Save Figure | Download Slide (.ppt)

The moving oscillator solution (solid line) for the SS beam and its moving force (dashed line) and quasi-static (dotted line) approximations

View Large | Save Figure | Download Slide (.ppt)

Approximations of the moving oscillator solution for the CC beam by the improved series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

View Large | Save Figure | Download Slide (.ppt)

A distributed system carrying a moving linear conservative oscillator

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Approximations of the moving oscillator solution for the CC beam by the ordinary series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

View Large | Save Figure | Download Slide (.ppt)

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Pesterev AV, Bergman LA. An Improved Series Expansion of the Solution to the Moving Oscillator Problem. ASME. J. Vib. Acoust. 1999;122(1):54-61. doi:10.1115/1.568436.

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An Improved Series Expansion of the Solution to the Moving Oscillator Problem

References

Figures

Displacement of the moving mass on the CC beam for v=2 m/s: (i) exact solution (solid line), (ii) moving force approximation (dashed line), and (iii) quasi-static approximation (dotted line).

Displacement of the moving mass on the CC beam for v=4 m/s: (i) exact solution (solid line), (ii) moving force approximation (dashed line), and (iii) quasi-static approximation (dotted line).

Displacement of the moving mass on the CC beam for v=24 m/s: (i) exact solution (solid line), (ii) moving force approximation (dashed line), and (iii) quasi-static approximation (dotted line).

The moving oscillator solution (solid line) for the CC beam and its moving force (dashed line) and quasi-static (dotted line) approximations

Shape of the string w(x,t) at t=0.5 for the moving oscillator problem (solid line) and its moving force (dashed line) and quasi-static (dotted line) approximations

Exact solution w(x,0.5) to the moving oscillator problem (solid line) (obtained by using 100 terms of the improved series) and two approximations to it obtained by means of five first terms of the improved series (dashed line) and five terms of the ordinary series (dotted line)

Approximations of the moving oscillator solution for the SS beam by the improved series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

Approximations of the moving oscillator solution for the SS beam by the ordinary series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

The moving oscillator solution (solid line) for the SS beam and its moving force (dashed line) and quasi-static (dotted line) approximations

Approximations of the moving oscillator solution for the CC beam by the improved series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

A distributed system carrying a moving linear conservative oscillator

Approximations of the moving oscillator solution for the CC beam by the ordinary series with 3 (dotted line), 6 (dashed line), and 12 (solid line) terms

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