Determination of the Mode Composition of Bending Vibrations of Plates Using Complex Spectral Analysis

Purpose. To investigate the complex spectra of bending oscillations (Lamb waves) with exponential temporal decay of the driving force amplitude for a rectangular plate with simply supported boundary conditions.

Methods. Complex spectral analysis of nonlinear interactions of optical, magnetic and acoustic waves in limited samples.

Results. Acoustic methods for examination of material structure and properties are now well investigated. By the analysis of Lamb waves propagation and scattering in various structures we may assume the presence of heterogeneities (layers, defects, etc.). But in some cases, such methods are not effective enough due to the complexity of the results interpretation – for example, resonance curves of several waves with different attenuations can’t be isolated. In the present study the complex frequency spectrum analysis was carried out for two vibration modes with equal frequencies and different attenuation factors. Scanning not only the real, but also the imaginary part of the amplitude-frequency spectrum allows one to determine the oscillation’s mode composition with a higher level of precision. The effect of complex resonance was previously considered for optical media, but the similarity between acoustic and electromagnetic wave equations allowed us to extend the complex spectroscopy principle to acoustics.

Conclusion. We suppose that application of complex spectral analysis of generated and recorded waves would significantly expand the potential of resonant acoustic methods. The complex spectrum allows one to distinguish normal modes with identical frequencies and different attenuation coefficients, besides the range of resonant curve along the axis of real frequency is bigger than one along the imaginary axis. This method can be applied to the investigation of non-linear interaction of magnetic, electro-magnetic and acoustic waves in limited medium. The complex spectral analysis can also be used for non-destructive testing and in seismology.  

1. Yukov E. A. Spektroskopiya [Spectroscopy]. Fisicheskaya entsiklopediya [Physical Encyclopedia]; ed. by A. M. Prohorov. Moscow, Bol'shaya Rossiiskaya entsiklopediya , 1994, vol. 4, р. 625.

2. Handbook of Vibrational Spectroscopy; ed. by J. Chalmers, P. Griffiths. 5 vol. set. Wiley, 2002. 4000 p.

3. Applied spectroscopy: a compact reference for practitioners; ed. by Jr. J. Workman, A. Springsteen. Academic Press, 1998. 539 p.

4. Skrabal P. M. Spectroscopy: An Interdisciplinary Integral Description of Spectroscopy from UV to NMR. Zürich, Vdf Hochschulverlag AG an der ETH, 2012.

5. Smith B. C. Fundamentals of Fourier transform infrared spectroscopy. 2th ed. CRC press, 2011. 207 p.

6. Bernath P. F. Infrared fourier transform emission spectroscopy. Chemical Society Reviews, 1996, vol. 25, no. 2, рр. 111–115.

7. Viktorov I. A. Zvukovye poverkhnostnye volny v tverdykh telakh [Sound superficial waves in solid bodies]. Moscow, Nauka Publ., 1981. 287 p.

8. Erofeyev V. I., Kazhayev V. V., Semerikova N. P. Volny v sterzhnyakh. Dispersiya. Dissipatsiya. Nelineinost' [Waves in cores. Dispersion. Dissipation. Nonlinearity]. Moscow, Fizmatlit Publ., 2002. 207 p.

9. Gorelik G. S. Kolebaniya i volny [Fluctuations and waves]. 2th ed. Moscow, Fizmatlit Publ., 1959. 572 p.

10. Elyashevich M. A. Spectroscopy of optical mixing and correlation of photons [Atomic and molecular spectroscopy]. Moscow, KomKniga Publ., 2006. 416 р.

11. Spectroscopy of optical mixing and correlation of photons; ed. by G. Cummings, E. Pike. Moscow, Mir Publ., 1978. 584 р.

12. Bessonov L. A. Teoreticheskie osnovy elektrotekhniki [Theoretical Foundations of Electrical Engineering]. Moscow, Vysshaya shkola Publ., 1996. 750 p.

13. Rosanov N. N. Kompleksnyi rezonans i spektroskopiya kompleksnykh chastot [Complex resonance and complex-frequency spectroscopy]. Pis'ma v zhurnal eksperimental'noi i teoreticheskoi fiziki = JETP Letters, 2009, vol. 90, no. 6, р. 428.

14. Rosanov N. N. Kompleksnyi rezonans pri frenelevskom otrazhenii impul'sov izlucheniya [Complex resonance in Fresnel reflection of radiation pulses]. Zhurnal eksperimental'noi i teoreticheskoi fiziki = Journal of Experimental and Theoretical Physics, 2010, vol. 111, no. 4, рр. 534–540.

15. Rosanov N. N. Kompleksnyi rezonans pri periodicheskoi raskachke ostsillyatora [Complex resonance at periodic excitation of an oscillator]. Optika i spektroskopiya = Optics and Spectroscopy, 2012, vol. 112, no. 6, рр. 898–901.

16. Rosanov N. N. Kompleksnyi rezonans i relaksatsionnye kolebaniya v lazere [Complex resonance and relaxation oscillations in a laser]. Optika i spektroskopiya = Optics and Spectroscopy, 2010, vol. 108, no. 4, рр. 651–655.

17. Rosanov N. N. Otklik mnogoprokhodnykh skhem na izluchenie s kompleksnoi chastotoi [Response of multipass schemes to radiation with complex frequency]. Optika i spektroskopiya = Optics and Spectroscopy, 2010, vol. 108, no. 4, рр. 634–636.

18. Rosanov N. N. Otrazhenie izlucheniya ot dvizhushchikhsya neodnorodnostei sredy: rezhimy neodnorodnykh voln i problema neodnoznachnosti dopplerovskikh chastotnykh sdvigov [Reflection of radiation from moving inhomogeneities of a medium: Inhomogeneous wave regimes and the ambiguity problem of doppler frequency shifts]. Optika i spektroskopiya = Optics and Spectroscopy, 2009, vol. 106, no. 4, рр. 609–613.

19. Lamb H. Dinamicheskaya teoriya zvuka [The dynamical theory of sound]. Moscow, Fizmatgiz Publ., 1960. 372 p.

20. Petrov Y. V., Gurevich S. Y., Golubev E. V. K teorii vozbuzhdeniya voln Lemba v metallakh impul'snym lazernym izlucheniem [Experimental determination of parameters of laser-generated lamb waves]. Akusticheskii zhurnal = Russian Journal of Nondestructive Testing, 2010, vol. 46, no. 3, рр. 185–188.

21. Kuzs’menko A. P., Zhukov E. A. Uprugie kolebaniya v plastinchatom obraztse ortoferrita ittriya, indutsirovannye dvizhushcheisya domennoi granitsei [Elastic oscillations induced by a moving domain wall in yttrium orthoferrite plate]. Pis'ma v zhurnal technichеskoi fiziki = Technical physics letters, 2006, vol. 32, no. 1, рр. 25–27.

22. Kuz’menko A. P., Zhukov E. A., Dobromyslov M. B. Excitation of bending vibration by a moving domain wall in a plate of yttrium orthoferrite. Journal of magnetism and magnetic materials, 2006, vol. 302, no. 2, рр. 436–438. doi: 10.1016/j.jmmm.2005.10.002

23. Kuz’menko A. P., Zhukov E. A., Dobromyslov M. B. Magneto-elastic resonant phenomena at the motion of the domain wall in weak ferromagnets. Journal of magnetism and magnetic materials, 2007, vol. 310, no. 2, рр. 1610–1612. doi: 10.1016/j.jmmm.2006.10.633

24. Zhukov E. A., Zhukova V. I., Kuz’menko A. P., Scherbakov Y. I. Nelineinye magnitoakusticheskie vzaimodeistviya v slabykh ferromagnetikakh [Nonlinear magnetoacoustic interactions in weak ferromagnets]. Izvestiya Rossiiskoi akademii nauk. Seriya fizicheskaya = Bulletin of the Russian Academy of Sciences: Physics, 2010, vol. 74, no. 10, рр. 1364–1366.

25. Petrov V. M., Srinivasan G. Theory of domain wall motion mediated magnetoelectric effects in a multiferroic composite. Physical Review B., 2014, vol. 90, no. 14, р. 144411.

26. Landau L. D., Lifshits E. M. Teoriya uprugosti [Theory of elasticity]. 2th ed. Moscow, Nauka Publ., 1965. 204 p.