Empirical Evidence for Incorrectness of Using Separation of Variables according to the Fourier Method in Oscillation Theory

The investigation concerns resonance properties. Separation of variables is the primary method of solving boundary value problems in oscillation theory. Separating variables means accepting the Fourier hypothesis that resonance at a given frequency involves all points on a given geometry oscillating at the same frequency (the resonant frequency). According to the Fourier hypothesis, the frequency response peaks of different sensors should lie on the same vertical line. Nastran and ANSYS use the Fourier hypothesis. We carried out frequency tests of items in the range of 10...100 Hz. Analysis of the test results showed (Fig. 2) that a resonance (at the same frequency) involved different points on the item geometry oscillating at different frequencies. We discovered frequency shifts measuring 1 Hz. Thus, the Fourier hypothesis does not work, and it is not correct to use separation of variables to solve the respective modal analysis problem.

References

[1] Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 1977. 736 p. (In Russ.).

[2] Collatz V.L. Eigenwertaufgaben mit technischen Anwendungen. Leipzig, Akademische Verlagsgesellschaft Geest & Portig K.-G., 1949. 466 p. (Russ. ed.: Zadachi na sobstvennye znacheniya. Moscow, Nauka Publ., 1968. 504 p.)

[3] Abgaryan K.A., Rapoport I.M. Dinamika raket [Rocket dynamics]. Moscow, Mashinostroenie Publ., 1969. 378 p. (In Russ.).

[4] Svetlitskiy V.A. Mekhanika gibkikh sterzhney i nitey [Mechanics of elastic rods and strands]. Moscow, Mashinostroenie Publ., 1978. 222 p. (In Russ.).

[5] Lyuminarskiy I.E. Raschet uprugikh sistem s odnostoronnimi svyazyami [Computation of elastic systems with one-way communication]. Moscow, Izd-vo MGIU Publ., 2006. 308 p. (In Russ.).

[6] Panovko Ya.G., Gubanova I.I. Ustoychivost’ i kolebaniya uprugikh system [Stability and oscillations of elastic systems]. Moscow, Nauka Publ., 1967. 420 p. (In Russ.).

[7] Rapoport I.M. Kolebaniya uprugoy obolochki, chastichno zapolnennoy zhidkost’yu [Oscillations of elastic shells, partially filled with fluid]. Moscow, Mashinostroenie Publ., 1967. 360 p. (In Russ.).

[8] Filippov A.P. Kolebaniya deformiruemykh system [Oscillations of deformable systems]. Moscow, Mashinostroenie Publ., 1970. 736 p. (In Russ.).

[9] Bolotin V.V. Sluchaynye kolebaniya uprugikh system [Random vibrations of elastic systems]. Moscow, Nauka Publ., 1979. 336 p. (In Russ.).

[10] Belyaev N.M., Ryadno A.A. Metody nestatsionarnoy teploprovodnosti [Methods of non-stationary heat transfer]. Moscow, Vysshaya shkola Publ., 1978. 328 p. (In Russ.).

[11] Dokuchaev L.V. Nelineynaya dinamika letatel’nykh apparatov s deformiruemymi elementami [Nonlinear aircraft dynamics with deformable elements]. Moscow, Mashinostroenie Publ., 1987. 232 p. (In Russ.).

[12] Heylen W., Lammens S., Sas P. Modal Analysis theory and testing. Katholieke Universiteit Leuven, 1998. 170 p. (Russ. ed.: Modal’nyy analiz. Teoriya i ispytaniya. Moscow, OOO Novatest Publ., 2010. 319 p.)

[13] Berns V.A., Dolgopolov A.V., Zhukov E.P. et al. Eksperimental’nyy modal’nyy analiz letatel’nykh apparatov [Experimental modal analysis of aircraft]. Novosibirsk, Izd-vo NGTU Publ., 2017. 160 p. (In Russ.).

[14] Sharma J.K. Theoretical and experimental modal analysis of beam. In: Engineering Vibration, Communication and Information Processing. Springer, 2019, pp. 177–186, doi: https://doi.org/10.1007/978-981-13-1642-5_16

[15] Kranjc T., Slavic J., Boltezar M. A comparison of the strain and the classic experimental modal analysis. J. Vib. Control, 2016, vol. 22, no. 2, pp. 371–381, doi: https://doi.org/10.1177/1077546314533137

[16] www.dataphysics.com: website of Data Physics corporation (accessed: 04.11.21).

[17] www.blms.ru: website of BLM Sinerzhi (accessed: 04.11.21). (In Russ.).

[18] Arinchev S.V. Newton’s third law is not a dogma but a computational hypothesis. Izvestiya Vysshikh uchebnykh zavedeniy. Mashinostroenie [BMSTU Journal of Mechanical Engineering], 2020, no. 6, pp. 36–50, doi: https://doi.org/10.18698/0536-1044-2020-6-36-50 (in Russ.).

[19] Arinchev S.V. Proektirovanie kosmicheskogo musorosborshchika [Design of space debris collector]. Moscow, SamIzdat Publ., 2020. (In Russ.).