An original method for evaluating the dissipative effect in SDOF systems due to the transient phenomenon was presented in a previous article. This method based on the use of an additional dimension, and the general relativity concept was validated experimentally. However, the function of the forcing frequency required to establish the metric of the space was identified using an experimental transfer function. In the present paper the main objective is to solve the geodesic equations in order to avoid the experimental identification of the function contained in the metric. The variational problem of the metric of Riemannian space gives three geodesic equations for the SDOF system studied. Solving these equations gives in particular the transient forced response which, when compared with experimental results, permits validating the proposed method and therefore proving that the transient motion bends the space-time.

Issue Section:
Technical Papers
Keywords:
damping, vibrations, differential geometry
Topics:
Damping, Tensors, Transients (Dynamics), Spacetime, Relativity (Physics)
1.
Lalanne
,
M.
,
2000
, “
Review: Modeling Damping in Mechanical Engineering Structures
,”
Shock and Vib.
,
7
, pp.
29
–
37
.
2.
Evan-Iwanowski, R. M., 1976, Resonance Oscillations in Mechanical Systems, Elsevier Scientific Publishing Company, Amsterdam.
3.
Mitropol’dki, Yu, A., 1964, Problems of the Asymptotic Theory of Nonstationary Vibrations, translated from Russian by Gutfreund, Ch., Nauka, Moscow.
4.
Nashif, A. H., Jones, D. I. G., and Henderson, J. P., 1985, Vibration Damping, J. Wiley & Sons, New York.
5.
Al Majid
,
A.
, and
Dufour
,
R.
,
2000
, “
An Event Dimension for Modeling Damping due to Time-Varying Forcing Frequency
,”
Nonlinear Dyn.
,
23
(
4
), pp.
303
–
318
.
6.
Chilov, G., 1975, Analyse Mathe´matique Fonctions de Plusieurs Variables Re´elles 1em´e et 2em´e parties, translated from Russian by Kharine, V. Mir Moscow.
7.
Einstein, A., 1971, Four Conferences on the Theory of Relativity Made to the University of Princeton, translated from German to English by M. Solovine, Gauthier-Villars, Paris.
8.
Einstein, A., 1971, The Theory of Relativity Restraint and General, Gauthier-Villars, Paris.
9.
Appelquist, T., Chodos, A., and Freund, P. D. O., 1987, Modern Kaluza-Klein Theories, Addison-Wesley Publishing Co.
10.
Adler, R., Bazin, M., and Schiffer, M., 1965, Introduction to General Relativity, McGraw-Hill.
11.
Fock, V., 1961, The Theory of Space, Time and Gravitation, translated from Russian by Kemmer, N., Pergamon Press.
12.
Elbaz, E., 1986, Relativite´ Ge´ne´rale et Gravitation, Edition Marketing.
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