On the passage through a resonance in a variable-mass mechanical system

The study of passage through resonance is an essential problem in many technical applications especially when analyzing the dynamics of above-resonance machines. In this case, it is important to determine the dynamic factor of transient resonant oscillations of a linear variable-mass oscillator under the action of an external force of variable frequency. In contrast to the existing literature, a new analytical solution to the Cauchy problem is found by quadratures for case when the disturbing force frequency and mass vary linearly. In this paper, the WKB approximation is used to describe transient resonance oscillations of a linearly varying mass under the action of an external force whose frequency varies linearly as well. The acceleration and coasting of the system with an increasing and decreasing oscillating mass are considered. The calculations showed that the resonance mode of a variable mass oscillator is not observed when the time-varying natural frequency coincides with the frequency of an external action but at a later time. The offset time decreases with increasing viscous forces. The obtained formulas make it possible to calculate the amplitudes of dangerous resonant oscillations.

References

[1] Katsuhiko Ogata. System Dynamics. Fourth Edition. University of Minnesota, 2005. 617 p.

[2] Ol’shanskii V.P., Ol’shanskii S.V. Metod VKB raschetakh nestatsionarnykh kolebanii ostsilliatorov [WKB method for unsteady oscillator]. Khar’kov, Mіs’kdruk publ., 2014. 264 p.

[3] Mitropol’skii Iu.A. Izbrannye trudy [SelectedWorks]. Kiev, Naukova dumka publ., vol. 2, 2012. 504 p.

[4] Cveticanin L. Dynamics of Machines with Variable Mass. Taylor & Francis Ltd, 1998. 300 p.

[5] Cveticanin L. A review on dynamics of mass variable systems. Journal of the Serbian Society for Computational Mechanics, 2012, vol. 6, no. 1, pp. 56–74.

[6] Ol’shanskii V.P., Ol’shanskii S.V. Modelirovanie kolebanii ostsi l l iatora l ineino-peremennoi massy pri impul ’snom nagruzhenii [Modeling oscillator linearly variable mass under impact loading]. Vіsnik NTU «KhPІ»: Matematichne modeliuvannia v tekhnі t s і ta tekhnologі iakh [Herald of NTU «KPI»: Mathematical modeling in engineering and technologies]. 2013, no. 37(1010), pp. 125—130.

[7] Karnakov B.M., Krainov V.P. WKB Approximation in Atomic Physics. Springer, 2013. 176 p.

[8] Zaitsev V.F., Polianin A.D. Spravochnik po obyknovennym di f ferentsial ’nym uravneni iam [Handbook of Differential Equations]. Moscow, FIZMATLIT publ., 2001. 576 p.