ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Degilevich E. A., Smirnov A. S. Oscillation frequencies of a chain pendulum in a model with weak nonlinearity. Izvestiya VUZ. Applied Nonlinear Dynamics, 2026, vol. 34, iss. 3, pp. 432-452. DOI: 10.18500/0869-6632-003214, EDN: VNVCZS

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
534.11
DOI: 
10.18500/0869-6632-003214
EDN: 
VNVCZS

Oscillation frequencies of a chain pendulum in a model with weak nonlinearity

Autors: 
Degilevich Egor Alekseevich, Institute of Problems of Mechanical Engineering of Russian Academy of Science
Smirnov Alexey Sergeevich, Peter the Great St. Petersburg Polytechnic University
Abstract: 

The purpose of this study is to derive an analytical solution for determining corrections to the oscillation frequencies of a chain pendulum for the initial distributed-parameter model and the rod finite-dimensional model with concentrated parameters, depending on the oscillation amplitude. Check the convergence of the solution from the finite-dimensional model to the solution from the distributed model as the number of segments that make up the model increases.

Methods. To describe the oscillatory motion of a chain pendulum in the presence of weak nonlinearity, an analytical approach based on asymptotic methods was used, including the harmonic balance equation. The finite-dimensional model is a rod system with an arbitrary number of inertial rods, pivotally connected to each other. As numerical experiments with a finite-dimensional model, simulations of its free oscillations were carried out using multibody dynamics methods, and the integration of the matrix equation of its motion was also carried out in the presence of collinear control, which allows the system to be accelerated according to the oscillation modes.

Results. Formulas are derived for calculating the oscillation frequencies of distributed model and finite-dimensional model of a chain pendulum, accounting for weak nonlinearity, in the dependence of the frequency number, oscillation amplitude, and the number of rods for the finite-dimensional model. The convergence of correction factors from the finite-dimensional model of a chain pendulum to the values of similar coefficients from the distributed model is demonstrated as the number of rods increases. Graphical illustrations of the calculation results are plotted for visual evaluation and comparison of the models.

Conclusion. Based on the obtained formulas and graphs, it was determined that the rod finite-dimensional model is suitable for describing the behavior of a chain pendulum in the presence of weak nonlinearity. In the case of system oscillations in this model at the lowest frequency, the order of ten rods is sufficient for the correct description and mathematical modeling of a chain pendulum. In the case of more complex motion according to the second or third oscillation mode, at least thirty rods are required to ensure sufficient smoothness of the oscillation modes.

Key words: 
chain pendulum
weak nonlinear
oscillation frequencies
distributed model
finite-dimensional model
multidimensional systems
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Received: 
11.01.2026
Accepted: 
25.01.2026
Available online: 
15.02.2026
Published: 
29.05.2026
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