Geodesic
Resonance in bounded nonlinear pendulum-type systems
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 3, pp. 289-296

Voir la notice de l'article provenant de la source Math-Net.Ru

Solving nonlinear differential equations with external forces is important for understanding resonant phenomena in the physics of oscillations. The article analyzes this problem basing on example of an ordinary second-order differential equation of the pendulum type, where the nonlinearity is described by a sinusoidal term. The phase plane of such an oscillator is constructed and its periodic trajectories are studied. It is illustrated that bounded nonlinearity matters only at intermediate amplitudes. The excitation of a nonlinear oscillator is carried out using a limited two–component force; the first its component corresponds to an oscillation at the resonant frequency of a linear oscillator, and the second is a limited function with a variable frequency. It is shown that with the appropriate choice of an external force, it is possible to obtain unlimited amplification of oscillations in a pendulum-type oscillator with amplitude linearly proportional to time. Spectral composition of the external force is investigated using short-time Fourier transform. It is demonstrated that in order to maintain the resonant mode, the frequency of the external force must continuously increase. Energy estimates of the external force and oscillator fluctuations depending on time are performed. The considered example is important for understanding resonant conditions in nonlinear problems.
Keywords: Nonlinear resonance, oscillator, mathematical pendulum
Mots-clés : short-time Fourier transform, spectrogram. 
@article{SVMO_2022_24_3_a2,
     author = {E. N. Pelinovsky and I. E. Melnikov},
     title = {Resonance in bounded nonlinear pendulum-type systems},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {289--296},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2022_24_3_a2/}
}
TY  - JOUR
AU  - E. N. Pelinovsky
AU  - I. E. Melnikov
TI  - Resonance in bounded nonlinear pendulum-type systems
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2022
SP  - 289
EP  - 296
VL  - 24
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2022_24_3_a2/
LA  - ru
ID  - SVMO_2022_24_3_a2
ER  - 
%0 Journal Article
%A E. N. Pelinovsky
%A I. E. Melnikov
%T Resonance in bounded nonlinear pendulum-type systems
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2022
%P 289-296
%V 24
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2022_24_3_a2/
%G ru
%F SVMO_2022_24_3_a2
E. N. Pelinovsky; I. E. Melnikov. Resonance in bounded nonlinear pendulum-type systems. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 3, pp. 289-296. http://geodesic.mathdoc.fr/item/SVMO_2022_24_3_a2/