Direct Numerical Procedure for Solution of Moving Oscillator Problems
Publication: Journal of Engineering Mechanics
Volume 126, Issue 5
Abstract
In this paper, the problem of a 1D elastic distributed system coupled with a moving linear oscillator, often referred to as the “moving oscillator” problem, is studied. The problem is formulated using a “relative displacement” model, which shows that, in the limiting case of infinite oscillator stiffness, the moving mass problem is recovered. The coupled equations of motion are recast into an integral equation that is amenable to solution by a direct numerical procedure. Both the integral equation and the numerical procedure show that the response of the elastic system at the current time depends only on the time history of its response at the positions of the oscillator. Numerical results are presented for the examples of a string and a simply supported beam and are compared to the moving force solutions. It is shown that the oscillator, with its stiffness suitably tuned, can excite the elastic structure into resonance.
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References
1.
Agrawal, O. P., and Saigal, S. ( 1986). “Dynamic responses of orthotropic plates under moving masses.” Recent trends in aeroelasticity, structures, and structural dynamics, P. Hajela, ed., University of Florida Press, Gainesville, Fla., 313–333.
2.
Alder, A. A., and Reismann, H. (1974). “Moving loads on an elastic plate strip.” J. Appl. Mech., 41, 713–718.
3.
Argento, A. (1995). “A spinning beam subjected to a moving deflection dependent load, Part I: Response and resonance.” J. Sound and Vibration, 182(4), 595–615.
4.
Benedetti, G. A. (1974). “Dynamic stability of a beam loaded by a sequence of moving mass particles.” J. Appl. Mech., 41, 1069–1071.
5.
Biggs, J. M. (1964). Introduction to structural dynamics. McGraw-Hill, New York.
6.
Bolotin, V. V. (1982). Application of the theory of probability and of the theory of reliability in the analysis of structures. Strojizdat, Moscow.
7.
Chen, C. H., and Wang, K. W. (1994). “An integrated approach toward the modeling and dynamic analysis of high-speed spindles, Part II: Dynamics under moving end load.” J. Vibration and Acoustics, 116, 514–522.
8.
Chonan, S. (1978). “Moving harmonic load on an elastically supported Timoshenko beam.” Z. Angew, Math, Mech., Berlin, 58, 9–15.
9.
Duffy, D. G. (1990). “The response of an infinite railroad track to a moving vibrating mass.” J. Appl. Mech., 57, 66–73.
10.
Felszeghy, S. F. (1996a). “The Timoshenko beam on an elastic foundation and subject to a moving step load, Part I: Steady-state response.” J. Vibration and Acoustics, 118, 277–284.
11.
Felszeghy, S. F. (1996b). “The Timoshenko beam on an elastic foundation and subject to a moving step load, Part II: Transient response.” J. Vibration and Acoustics, 118, 285–291.
12.
Fryba, L. (1957). “Infinite beam on an elastic foundation subjected to a moving load.” Aplikace Matematiky, Prague, 2(2), 105–132.
13.
Fryba, L. (1972). Vibration of solids and structures under moving loads. Noordhoff, Groningen, The Netherlands.
14.
Fryba, L. (1976). “Non-stationary response of a beam to moving random force.” J. Sound and Vibration, 46(3), 323–338.
15.
Fryba, L. (1977). “Stationary response of a beam to a moving continuous random load.” Acta Technica Czech Science Advanced Views, 22(4), 444–479.
16.
Fryba, L. ( 1986). “Random vibration of a beam on elastic foundation under moving force.” Random vibration-status and recent developments, I. Elishakoff and R. H. Lyon, eds., Elsevier Science, New York, 127–147.
17.
Fujino, Y., Bhartia, B., and Kasahara, S. (1994). “Active control of traffic-induced vibrations in highway bridges.” Proc., 1st World Conf. on Struct. Control, G. W. Housner, S. F. Masri, and A. G. Chassiakos, eds., University of Southern California, Los Angeles, FA1-3–FA1-13.
18.
Henchi, K., Fafard, M., Dhatt, G., and Talbot, M. (1997). “Dynamic behaviour of multi-span beams under moving loads.” J. Sound and Vibration, 199(1), 33–50.
19.
Henchi, K., Fafard, M., and Talbot, M. (1998a). “Dynamic analysis of road bridges for bridge-vehicles. I. Numerical aspects.” Can. J. Civ. Engrg., Ottawa, 25(1), 161–173.
20.
Henchi, K., Fafard, M., Talbot, M., and Dhatt, G. (1998b). “An efficient algorithm for dynamic analysis of bridges under moving vehicles using a coupled modal and physical components approach.” J. Sound and Vibration, 212(4), 663–683.
21.
Henchi, K., Talbot, M., and Fafard, M. (1998c). “Dynamic analysis of road bridges for bridge vehicles. II. Application to Senneterre Bridge in Quebec.” Can. J. Civ. Engrg., Ottawa, 25(1), 174–178.
22.
Huang, S. C., and Chiou, W. J. (1997). “Modeling and vibration analysis of spinning-disk and moving-head assembly in computer storage systems.” J. Vibration and Acoustics, 119(2), 185–191.
23.
Huang, S. C., and Hsu, B. S. (1990). “Resonant phenomena of a rotating cylindrical shell subjected to a harmonic moving load.” J. Sound and Vibration, 136(2), 215–228.
24.
Inglis, C. E. (1934). A mathematical treatise on vibrations in railway bridges. Cambridge University Press, Cambridge, U.K.
25.
Iwan, W. D., and Stahl, K. J. (1973). “The response of an elastic disk with a moving mass system.” J. Appl. Mech., 40, 445–451.
26.
Johnson, E. A. ( 1993). “Firing-induced vibration of an electromagnetic railgun rail,” MS thesis, University of Illinois, Urbana, Ill.
27.
Kenny, J. T. (1954). “Steady-state vibrations of beam on elastic foundation for moving load.” J. Appl. Mech., 21, 359–364.
28.
Kerr, A. D. (1972). “The continuously supported rail subjected to an axial force and a moving load.” Int. J. Mech. Sci., 14, 71–78.
29.
Knowles, J. K. (1968). “On the dynamic response of a beam to a randomly moving load.” J. Appl. Mech., 35, 1–6.
30.
Knowles, J. K. (1970). “A note on the response of a beam to a randomly moving force.” J. Appl. Mech., 37, 1192–1194.
31.
Lee, U. (1996). “Revisiting the moving mass problem: Onset of separation between the mass and beam.” J. Vibration and Acoustics, 118, 516–521.
32.
Lin, Y.-H., and Trethewey, M. W. (1990). “Finite element analysis of elastic beams subjected to moving dynamic loads.” J. Sound and Vibration, 136(2), 323–342.
33.
Mackertich, S. (1990). “Moving load on a Timoshenko beam.” J. Acoustical Soc. of Am., 88, 1175–1178.
34.
Mathews, P. M. (1958). “Vibrations of a beam on elastic foundation.” Z. Angew, Math, Mech., Berlin, 38, 105–115.
35.
Mote, C. D., Jr. (1977). “Moving-load stability of a circular plate on a floating central collar.” J. Acoustical Soc. of Am., 61, 439–447.
36.
Pesterev, A. V., and Bergman, L. A. (1997a). “Response of elastic continuum carrying moving linear oscillator.”J. Engrg. Mech., ASCE, 123(8), 878–884.
37.
Pesterev, A. V., and Bergman, L. A. (1997b). “Vibration of elastic continuum carrying accelerating oscillator.”J. Engrg. Mech., ASCE, 123(8), 886–889.
38.
Pesterev, A. V., and Bergman, L. A. (1998a). “A contribution to the moving mass problem.” J. Vibration and Acoustics, 120, 824–826.
39.
Pesterev, A. V., and Bergman, L. A. (1998b). “Response of a nonconservative continuous system to a moving concentrated load.” J. Appl. Mech., 65, 436–444.
40.
Sadiku, S., and Leipholz, H. H. E. (1987). “On the dynamics of elastic systems with moving concentrated masses.” Ingenieur-Archiv, Berlin, 57, 223–242.
41.
Saigal, S. (1986). “Dynamic behavior of beam structures carrying moving masses.” J. Appl. Mech., 53, 222–224.
42.
Saigal, S., Agrawal, O. P., and Stanisic, M. M. (1987). “Influence of moving masses on rectangular plate dynamics.” Ingenieur-Archiv, Berlin, 57, 187–196.
43.
Stanisic, M. M. (1985). “On a new theory of the dynamic behavior of the structures carrying moving masses.” Ingenieur-Archiv, Berlin, 55, 176–185.
44.
Stanisic, M. M., Euler, J. A., and Montgomery, S. T. (1974). “On a theory concerning the dynamical behavior of structures carrying moving masses.” Ingenieur-Archiv, Berlin, 43, 295–305.
45.
Stanisic, M. M., and Hardin, J. C. (1969). “On response of beams to an arbitrary number of moving masses.” J. Franklin Inst., 287, 115–123.
46.
Stokes, G. G. (1883). “Discussions of a differential equation relating to the breaking of railway bridges.” Math. and Phys. Papers, 2, 178–220.
47.
Tan, C. P., and Shore, S. (1968). “Response of horizontally curved bridge to moving load.”J. Struct. Div., ASCE, 94(12), 2135–2151.
48.
Timoshenko, S., Young, D. H., and Weaver, W. (1974). Vibration problems in engineering. Wiley, New York.
49.
Ting, E. C., and Yener, M. (1983). “Vehicle-structure interactions in bridge dynamics.” Shock and Vibration Dig., 15(12), 3–9.
50.
Weisenel, G. N., and Schlack, A. L., Jr. (1993). “Response of annular plates to circumferentially and radially moving loads.” J. Appl. Mech., 60, 649–661.
51.
Yu, R. C., and Mote, C. D., Jr. (1987). “Vibration and parametric excitation in asymmetric circular plates under moving loads.” J. Sound and Vibration, 119(3), 409–427.
52.
Zhu, W. D., and Mote, C. D., Jr. (1994). “Free and forced response of an axially moving string transporting a damped linear oscillator.” J. Sound and Vibration, 177(5), 591–610.
Information & Authors
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Published In
Journal of Engineering Mechanics
Volume 126 • Issue 5 • May 2000
Pages: 462 - 469
History
Received: Mar 18, 1999
Published online: May 1, 2000
Published in print: May 2000
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