Invariant Characteristics of Forced Oscillations of a Beam with Longitudinal Compression

Oscillations of an elastic beam with longitudinal compression are considered. The beam consists of two steel strips connected on free ends and fixed on opposite ones. Compression is achieved by a strained string. Excitation of oscillations is performed by exposure of alternating magnetic field on a magnet placed on the loose end. The law of motion with a change in the frequency of the harmonic action is registered. As a result of the full-scale experiment a large set of data is obtained. This set contains ordered periodic oscillations as well as disordered oscillations specific to dynamical systems with chaotic behaviour. To study the invariant numerical characteristics of the attractor of the corresponding dynamical system, a correlation integral and a correlation dimensionality as well as β-statentropy are calculated. A large numerical experiment showed that the calculation of β-statentropy is preferable to the calculation of the correlation index. Based on the developed algorithms the dependence of β- statentropy on the frequency of the external action is constructed. The constructed dependence can serve as an effective tool for measuring the adequacy of the mathematical model of forced oscillations of buckling beam driven oscillations.

1. Moon F.C. and Holmes P.J., “A magnetoelastic strange attractor”, Journal of Sound and Vibration, 65:2 (1979), 275 – 296.

2. Tam JeeIan, Holmes Philip, “Revisiting a magneto-elastic strange attractor”, Journal of Sound and Vibration, 333:6 (2014), 1767–1780.

3. Guckenheimer J. and Holmes P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer, 1983.

4. Kumar K.Aravind, Ali Shaikh Faruque, Arockiarajan A., “Magneto-elastic oscillator: Modeling and analysis with nonlinear magnetic interaction”, Journal of Sound and Vibration, 393 (2017), 265–284.

5. Timofeev E.A., “Invariants of measures admiting statistical estimates”, St. Petersburg Math. J., 17:3 (2006), 527–551.

6. Yoshisuke Ueda, “Randomly transitional phenomena in the system governed by Duffing’s equation”, Nagoya University Institute of Plasma Physics, Japan, 1978, №IPPJ–341.

7. Glyzin D. S., Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “The Dynamic Renormalization Method for Finding the Maximum Lyapunov Exponent of a Chaotic Attractor”, Differ. Equ., 41:2 (2005), 284–289.

8. Smirnov L.V., Vyvod uravneniy dinamiki uprugikh sistem, Uchebnoe posobie, UNN, Nizhniy Novgorod, 1997, 15 pp., (in Russian)

9. Kapitanov D.V., Ovchinnikov V.F., Smirnov L.V., “The dynamics of an axially loaded elastic bar after loss of stability”, Problems of strength and plasticity, 76:3 (2014), 205–216, (in Russian)

10. Saranin V.A., “About chaotic behaviour of an electrostatic pendulum at parametrical influence”, Bulletin of Perm University. Series: Physics, 27–28 (2014), 18–23 (in Russian)

11. M.V. Lokhanin, Ju.V. Shibalova, “Buckling beam as nonvolatile memory cell”, Vestnik Yaroslavskogo gosudarstvennogo universiteta im. P.G. Demidova. Seriya Yestestvennyye i tekhnicheskiye nauki, 2014, №1, 34–37 (in Russian)